Triangular numbers

A triangular number is a count of objects arranged into an equilateral triangle. Much like how a square number is a count of objects arranged into a square.

The nth triangular number is the sum of the first n non-negative integers. $T_{n}=\sum _{k=1}^{n}k=1+2+3+\dotsb +n$

Triangular numbers may be calculated by the explicit formulas: $T_{n}={\frac {n(n+1)}{2}}={n+1 \choose 2}$

where $\textstyle {n+1 \choose 2}$ is the binomial coefficient "n plus one choose two".

Analogous to square roots, we may also calculate a triangular root. Numbers that have an integer triangular root are triangular numbers.

The real triangular root of a number x may be found using: $n={\frac {{\sqrt {8x+1}}-1}{2}}$

Similar to how cubic numbers are square numbers extended into a third dimension, triangular numbers extended into a third dimension are known as tetrahedral numbers.

The nth tetrahedral number is the sum of the first n triangular numbers. $Te_{n}=\sum _{k=1}^{n}T_{k}$

Or, may be calculated directly: $Te_{n}={\frac {n(n+1)(n+2)}{6}}={\frac {n^{\overline {3}}}{3!}}={\binom {n+2}{3}}$ (Binomial "n plus two choose three".)

One may find the real tetrahedral root of x using the formula: $n={\sqrt[{3}]{3x+{\sqrt {9{x^{2}}-{\frac {1}{27}}}}}}+{\sqrt[{3}]{3x-{\sqrt {9{x^{2}}-{\frac {1}{27}}}}}}-1$
Depending on the math precision of your particular language, may need to be rounded to the nearest 1e-16 or so.

Extending into a fourth dimension we get pentatopic numbers.

Again, the nth pentatope is the sum of the first n tetrahedral numbers, $P_{n}=\sum _{k=1}^{n}Te_{k}$

or explicitly: $P_{n}={\frac {n(n+1)(n+2)(n+3)}{24}}={\binom {n+3}{4}}$ (Binomial "n plus three choose four".)

The pentatopic real root of x may be found using: $n={\frac {{\sqrt {5+4{\sqrt {24x+1}}}}-3}{2}}.$

In general, these all belong to the class figurate numbers as they are based on r dimensional geometric figures. Sometimes they are referred to as r-simplex numbers. In geometry a simplex is the simplest possible r-dimensional object.

You may easily extend to an arbitrary dimension r using binomials. Each term n in dimension r is $r_{n}={\binom {n+r-1}{r}}$

There is no known general formula to find higher r-simplex roots.

Task

Find and display the first 30 triangular numbers (r = 2).
Find and display the first 30 tetrahedral numbers (r = 3).
Find and display the first 30 pentatopic numbers (r = 4).
Find and display the first 30 12-simplex numbers (r = 12).
Find and display the triangular root, the tetrahedral root, and the pentatopic root for the integers:
- 7140
- 21408696
- 26728085384
- 14545501785001

See also

Wikipedia: Triangular numbers Wikipedia: Tetrahedral numbers Wikipedia: Pentatopic numbers Wikipedia: Figurate numbers Wikipedia: Simplex(geometry) OEIS:A000217 - Triangular numbers: a(n) = binomial(n+1,2) OEIS:A000292 - Tetrahedral numbers: a(n) = binomial(n+2,3) OEIS:A000332 - Pentatope numbers: a(n) = binomial(n+3,4) Related task: Evaluate binomial coefficients Related task: Pascal's triangle

ALGOL 68

Assumes LONG INT is at least 64 bits.
Some roots that should be integers have non-integral values when calculated by the formulae in the task. To get the correct values (if possible), the root procedures used here calculate float value and then use the nearest integer or nearest integer + 1 if they are exact.

BEGIN # show some triangular, tetrahedral, ... numbers and roots             #
    # prints a row of LONG INTs, with a title, newlines after every per-line #
    # and each number in the specified width                                 #
    PROC show values = ( STRING title, []LONG INT v, INT per line, INT width )VOID:
         BEGIN
            print( ( title, ":", newline ) );
            INT on line := 0;
            FOR i FROM LWB v TO UPB v DO
                print( ( whole( v[ i ], -width ) ) );
                IF ( on line +:= 1 ) = per line THEN
                    print( ( newline ) );
                    on line := 0
                FI
            OD;
            IF on line /= 0 THEN print( ( newline ) ) FI;
            print( ( newline ) )
         END # show values # ;

    # calculate the first 30 triangular, tetrahedral, etc. numbers           #
    [ 1 : 30 ]LONG INT triangular;  triangular[  1 ] := 0;
    [ 1 : 30 ]LONG INT tetrahedral; tetrahedral[ 1 ] := 0;
    [ 1 : 30 ]LONG INT pentatopic;  pentatopic[  1 ] := 0;
    [ 1 : 30 ]LONG INT simplex12;   simplex12[   1 ] := 0;
    FOR i FROM 2 TO 30 DO
        triangular[  i ] := triangular[  i - 1 ] + i - 1;
        tetrahedral[ i ] := tetrahedral[ i - 1 ] + triangular[  i ];
        pentatopic[  i ] := pentatopic[  i - 1 ] + tetrahedral[ i ];
        simplex12[   i ] := pentatopic[  i     ]
    OD;
    FROM 5 TO 12 DO
        FOR i FROM 2 TO 30 DO
            simplex12[ i ] +:= simplex12[ i - 1 ]
        OD
    OD;
    show values( "First 30 Triangular  numbers", triangular,  6,  4 );
    show values( "First 30 Tetrahedral numbers", tetrahedral, 6,  5 );
    show values( "First 30 Pentatopic  numbers", pentatopic,  6,  6 );
    show values( "First 30 12-Simplex  numbers", simplex12,   6, 12 );

    # show some triangular, tetrahedral, etc. roots                          #

    # returns the cube root of x                                             #
    PROC long crt = ( LONG REAL x )LONG REAL: long exp( long ln( x ) / 3 );
    # returns a LONG REAL approximation to the  triangular root of x         #
    PROC  real triangular root = ( LONG INT x )LONG REAL: ( long sqrt( ( 8 * x ) + 1 ) - 1 ) / 2;
    # returns a LONG REAL approximation to the tetrahedral root of x         #
    PROC real tetrahedral root = ( LONG INT x )LONG REAL:
         BEGIN
             LONG REAL t = long sqrt( ( 9 * x * x ) - ( 1 / 27 ) ); 
             long crt( ( 3 * x ) + t ) + long crt( ( 3 * x ) - t ) - 1
         END # tetrahedral root # ;
    # returns a LONG REAL approximation to the  pentatopic root of x         #
    PROC  real pentatopic root = ( LONG INT x )LONG REAL:
         ( long sqrt( 5 + ( 4 * long sqrt( ( 24 * x ) + 1 ) ) ) - 3 ) / 2;
    # returns an integer root of x, if the approximation (possibly + 1 ) = x #
    #                               the approximation otherwise              #
    PROC try integer root = ( LONG INT x, LONG REAL real root, PROC( LONG INT )LONG INT f )LONG REAL:
         IF LONG INT ir = ENTIER real root;
            f( ir ) = x
         THEN ir
         ELIF f( ir + 1 ) = x
         THEN ir + 1
         ELSE real root
         FI # try integer root # ;
    # returns the  triangular root of x                                      #
    PROC  triangular root = ( LONG INT x )LONG REAL:
         try integer root( x
                         , real triangular root( x )
                         , ( LONG INT n )LONG INT: ( n * ( n + 1 ) ) OVER 2
                         );
    # returns the tetrahedral root of x                                      #
    PROC tetrahedral root = ( LONG INT x )LONG REAL:
         try integer root( x
                         , real tetrahedral root( x )
                         , ( LONG INT n )LONG INT: ( n * ( n + 1 ) * ( n + 2 ) ) OVER 6
                         );
    # returns the  pentatopic root of x                                      #
    PROC  pentatopic root = ( LONG INT x )LONG REAL:
         try integer root( x
                         , real pentatopic root( x )
                         , ( LONG INT n )LONG INT: ( n * ( n + 1 ) * ( n + 2 ) * ( n + 3 ) ) OVER 24
                         );
    []LONG INT root test = ( 7140, 21408696, 26728085384, 14545501785001 );
    FOR i FROM LWB root test TO UPB root test DO
        PROC show = ( LONG REAL x )STRING:
             IF ENTIER x = x THEN whole( x, -6 ) + "      " ELSE fixed( x, -12, 5 ) FI;
        print( ( "Roots of ", whole( root test[ i ], 0 ), newline, "   " ) );
        print( ( " triangular: ",  show(  triangular root( root test[ i ] ) ) ) );
        print( ( " tetrahedral: ", show( tetrahedral root( root test[ i ] ) ) ) );
        print( ( " pentatopic: ",  show(  pentatopic root( root test[ i ] ) ) ) );
        print( ( newline ) )
    OD
END

Output:

First 30 Triangular  numbers:
   0   1   3   6  10  15
  21  28  36  45  55  66
  78  91 105 120 136 153
 171 190 210 231 253 276
 300 325 351 378 406 435

First 30 Tetrahedral numbers:
    0    1    4   10   20   35
   56   84  120  165  220  286
  364  455  560  680  816  969
 1140 1330 1540 1771 2024 2300
 2600 2925 3276 3654 4060 4495

First 30 Pentatopic  numbers:
     0     1     5    15    35    70
   126   210   330   495   715  1001
  1365  1820  2380  3060  3876  4845
  5985  7315  8855 10626 12650 14950
 17550 20475 23751 27405 31465 35960

First 30 12-Simplex  numbers:
           0           1          13          91         455        1820
        6188       18564       50388      125970      293930      646646
     1352078     2704156     5200300     9657700    17383860    30421755
    51895935    86493225   141120525   225792840   354817320   548354040
   834451800  1251677700  1852482996  2707475148  3910797436  5586853480

Roots of 7140
    triangular:    119       tetrahedral:     34       pentatopic:     18.87665
Roots of 21408696
    triangular:   6543       tetrahedral:    503.56183 pentatopic:    149.06095
Roots of 26728085384
    triangular: 231205.40557 tetrahedral:   5432       pentatopic:    893.44246
Roots of 14545501785001
    triangular: 5393607.1581 tetrahedral:  44355.77738 pentatopic:   4321

AppleScript

on rSimplexNumber(r, n)
    set n to n - 1 -- "nth" is 0-based in the formula!
    set numerator to n
    set denominator to 1
    repeat with dimension from 2 to r
        set numerator to numerator * (n + dimension - 1)
        set denominator to denominator * dimension
    end repeat
    
    return numerator div denominator
end rSimplexNumber

on triangularRoot(x)
    return ((8 * x + 1) ^ 0.5 - 1) / 2
end triangularRoot

on tetrahedralRoot(x)
    -- NOT (((9 * (x ^ 2) - 1 / 27) ^ 0.5 + 3 * x) ^ (1 / 3)) * 2 - 1  !
    return (((9 * (x ^ 2) - 1 / 27) ^ 0.5 + 3 * x) ^ (1 / 3)) - 1
end tetrahedralRoot

on pentatopicRoot(x)
    return (((24 * x + 1) ^ 0.5 * 4 + 5) ^ 0.5 - 3) / 2
end pentatopicRoot

on intToText(int)
    set txt to ""
    repeat while (int > 99999999)
        set txt to ((100000000 + int mod 100000000) as integer as text)'s text 2 thru 9 & txt
        set int to int div 100000000
    end repeat
    return (int as text) & txt
end intToText

on join(lst, delim)
    set astid to AppleScript's text item delimiters
    set AppleScript's text item delimiters to delim
    set txt to lst as text
    set AppleScript's text item delimiters to astid
    return txt
end join

on task()
    set output to {}
    set padding to "               "
    set columnWidth to (count intToText(rSimplexNumber(12, 30))) + 2
    repeat with rt in {{2, "triangular"}, {3, "tetrahedral"}, {5, "pentatopic"}, {12, "12-simplex"}}
        set {r, type} to rt
        set end of output to linefeed & "First thirty " & type & " numbers:"
        set these6 to {}
        repeat with n from 1 to 30
            set this to intToText(rSimplexNumber(r, n))
            set these6's end to (padding & this)'s text -columnWidth thru -1
            if (n mod 6 = 0) then
                set end of output to join(these6, "")
                set these6 to {}
            end if
        end repeat
    end repeat
    repeat with n in {7140, 21408696, 2.6728085384E+10, 1.4545501785001E+13}
        set end of output to linefeed & "Roots of " & intToText(n) & ":"
        set end of output to "  Triangular root: " & triangularRoot(n)
        set end of output to "  Tetrahedral root: " & tetrahedralRoot(n)
        set end of output to "  Pentatopic root: " & pentatopicRoot(n)
    end repeat
    return join(output, linefeed)
end task

return task()

Output:

"
First thirty triangular numbers:
           0           1           3           6          10          15
          21          28          36          45          55          66
          78          91         105         120         136         153
         171         190         210         231         253         276
         300         325         351         378         406         435

First thirty tetrahedral numbers:
           0           1           4          10          20          35
          56          84         120         165         220         286
         364         455         560         680         816         969
        1140        1330        1540        1771        2024        2300
        2600        2925        3276        3654        4060        4495

First thirty pentatopic numbers:
           0           1           6          21          56         126
         252         462         792        1287        2002        3003
        4368        6188        8568       11628       15504       20349
       26334       33649       42504       53130       65780       80730
       98280      118755      142506      169911      201376      237336

First thirty 12-simplex numbers:
           0           1          13          91         455        1820
        6188       18564       50388      125970      293930      646646
     1352078     2704156     5200300     9657700    17383860    30421755
    51895935    86493225   141120525   225792840   354817320   548354040
   834451800  1251677700  1852482996  2707475148  3910797436  5586853480

Roots of 7140:
  Triangular root: 119.0
  Tetrahedral root: 33.990473597552
  Pentatopic root: 18.876646615928

Roots of 21408696:
  Triangular root: 6543.0
  Tetrahedral root: 503.561166334548
  Pentatopic root: 149.060947375266

Roots of 26728085384:
  Triangular root: 2.312054055653E+5
  Tetrahedral root: 5431.99993864654
  Pentatopic root: 893.442456751685

Roots of 14545501785001:
  Triangular root: 5.393607158145E+6
  Tetrahedral root: 4.435577737656E+4
  Pentatopic root: 4321.0"

FreeBASIC

Translation of: Wren

Dim As Integer n, r, t(0 To 30)
t(0) = 0

Print "The first 30 triangular numbers are:"
For n = 1 To 30
    t(n) = t(n-1) + n - 1
    If n Mod 6 = 0 Then Print Using "####"; t(n) Else Print Using "####"; t(n);
Next n

Print !"\nThe first 30 tetrahedral numbers are:"
For n = 1 To 30 
    t(n) += t(n-1)
    Print Using "#####"; t(n);
    If n Mod 6 = 0 Then Print
Next n

Print !"\nThe first 30 pentatopic numbers are:"
For n = 1 To 30
    t(n) += t(n-1)
    Print Using "######"; t(n);
    If n Mod 6 = 0 Then Print
Next n

Print !"\nThe first 30 12-simplex numbers are:"
For r = 5 To 12
    For n = 1 To 30
        t(n) += t(n-1)
        If r = 12 Then 
            Print Using "###########"; t(n);
            If n Mod 6 = 0 Then Print
        End If
    Next n
Next r

#define cRec27 1/sqr(27)
Dim As Integer xs(1 To 4) = {7140, 21408696, 26728085384, 14545501785001}
Dim As Double x, y, z

For i As Byte = 1 To 4
    z = xs(i)
    Print !"\nRoots of"; xs(i); ":"
    Print " triangular:"; (Sqr(8*z+1)-1)/2
    
    y = 3*z  
    x = Sqr((y-cRec27)*(y+cRec27))
    Print "tetrahedral:"; Iif(x < y, Exp(Log(y+x)/3)+Exp(Log(y-x)/3)-1, Exp(Log(6)/3)*Exp(Log(z)/3)-1)
    
    Print " pentatopic:"; (Sqr(5+4*Sqr(24*z+1))-3)/2
Next i
Sleep

Output:

The first 30 triangular numbers are:
   0   1   3   6  10  15
  21  28  36  45  55  66
  78  91 105 120 136 153
 171 190 210 231 253 276
 300 325 351 378 406 435

The first 30 tetrahedral numbers are:
    0    1    4   10   20   35
   56   84  120  165  220  286
  364  455  560  680  816  969
 1140 1330 1540 1771 2024 2300
 2600 2925 3276 3654 4060 4495

The first 30 pentatopic numbers are:
     0     1     5    15    35    70
   126   210   330   495   715  1001
  1365  1820  2380  3060  3876  4845
  5985  7315  8855 10626 12650 14950
 17550 20475 23751 27405 31465 35960

The first 30 12-simplex numbers are:
          0          1         13         91        455       1820
       6188      18564      50388     125970     293930     646646
    1352078    2704156    5200300    9657700   17383860   30421755
   51895935   86493225  141120525  225792840  354817320  548354040
  834451800 1251677700 1852482996 2707475148 3910797436 5586853480

Roots of 7140:
 triangular: 119
tetrahedral: 34.00000000179027
 pentatopic: 18.87664661592801

Roots of 21408696:
 triangular: 6543
tetrahedral: 503.5611663345483
 pentatopic: 149.0609473752659

Roots of 26728085384:
 triangular: 231205.4055652559
tetrahedral: 5431.99993864654
 pentatopic: 893.4424567516849

Roots of 14545501785001:
 triangular: 5393607.158145173
tetrahedral: 44355.77737655847
 pentatopic: 4321

Go

Translation of: Wren

Library: Go-rcu

Library: bigfloat

I've had to use a third party library to calculate cube roots as the big.Float type in the standard library doesn't have a function for this. The results (to 24 d.p) are the same as the Raku example with the exception of the tetrahedral root for the largest integer which differs in the last three places.

package main

import (
    "fmt"
    "github.com/ALTree/bigfloat"
    "math/big"
    "rcu"
)

func main() {
    t := make([]int, 30)
    for n := 1; n < 30; n++ {
        t[n] = t[n-1] + n
    }
    fmt.Println("The first 30 triangular numbers are:")
    rcu.PrintTable(t, 6, 3, false)

    for n := 1; n < 30; n++ {
        t[n] += t[n-1]
    }
    fmt.Println("\nThe first 30 tetrahedral numbers are:")
    rcu.PrintTable(t, 6, 4, false)

    for n := 1; n < 30; n++ {
        t[n] += t[n-1]
    }
    fmt.Println("\nThe first 30 pentatopic numbers are:")
    rcu.PrintTable(t, 6, 5, false)

    for r := 5; r <= 12; r++ {
        for n := 1; n < 30; n++ {
            t[n] += t[n-1]
        }
    }
    fmt.Println("\nThe first 30 12-simplex numbers are:")
    rcu.PrintTable(t, 6, 10, false)

    const prec = 256
    xs := []float64{7140, 21408696, 26728085384, 14545501785001}
    root := new(big.Float)
    temp := new(big.Float)
    temp2 := new(big.Float)
    one := big.NewFloat(1)
    two := big.NewFloat(2)
    three := big.NewFloat(3)
    four := big.NewFloat(4)
    five := big.NewFloat(5)
    eight := big.NewFloat(8)
    nine := big.NewFloat(9)
    twentyFour := big.NewFloat(24)
    twentySeven := big.NewFloat(27)
    third := new(big.Float).SetPrec(prec).Quo(one, three)
    for _, x := range xs {
        bx := big.NewFloat(x).SetPrec(prec)
        fmt.Printf("\nRoots of %d:\n", int(x))
        root.Mul(bx, eight)
        root.Add(root, one)
        root.Sqrt(root)
        root.Sub(root, one)
        root.Quo(root, two)
        fmt.Printf("%14s: %.24f\n", "triangular", root)

        temp.Mul(bx, bx)
        temp.Mul(temp, nine)
        temp.Sub(temp, new(big.Float).SetPrec(prec).Quo(one, twentySeven))
        temp.Sqrt(temp)
        temp2.Mul(bx, three)
        temp2.Sub(temp2, temp)
        temp2 = bigfloat.Pow(temp2, third)
        root.Mul(bx, three)
        root.Add(root, temp)
        root = bigfloat.Pow(root, third)
        root.Add(root, temp2)
        root.Sub(root, one)
        fmt.Printf("%14s: %.24f\n", "tetrahedral", root)

        root.Mul(bx, twentyFour)
        root.Add(root, one)
        root.Sqrt(root)
        root.Mul(root, four)
        root.Add(root, five)
        root.Sqrt(root)
        root.Sub(root, three)
        root.Quo(root, two)
        fmt.Printf("%14s: %.24f\n", "pentatonic", root)
    }
}

Output:

The first 30 triangular numbers are:
  0   1   3   6  10  15 
 21  28  36  45  55  66 
 78  91 105 120 136 153 
171 190 210 231 253 276 
300 325 351 378 406 435 

The first 30 tetrahedral numbers are:
   0    1    4   10   20   35 
  56   84  120  165  220  286 
 364  455  560  680  816  969 
1140 1330 1540 1771 2024 2300 
2600 2925 3276 3654 4060 4495 

The first 30 pentatopic numbers are:
    0     1     5    15    35    70 
  126   210   330   495   715  1001 
 1365  1820  2380  3060  3876  4845 
 5985  7315  8855 10626 12650 14950 
17550 20475 23751 27405 31465 35960 

The first 30 12-simplex numbers are:
         0          1         13         91        455       1820 
      6188      18564      50388     125970     293930     646646 
   1352078    2704156    5200300    9657700   17383860   30421755 
  51895935   86493225  141120525  225792840  354817320  548354040 
 834451800 1251677700 1852482996 2707475148 3910797436 5586853480 

Roots of 7140:
    triangular: 119.000000000000000000000000
   tetrahedral: 34.000000000000000000000000
    pentatonic: 18.876646615928006607901783

Roots of 21408696:
    triangular: 6543.000000000000000000000000
   tetrahedral: 503.561826974636514048196130
    pentatonic: 149.060947375265867484387575

Roots of 26728085384:
    triangular: 231205.405565255836957291031961
   tetrahedral: 5432.000000000000000000000000
    pentatonic: 893.442456751684869888466212

Roots of 14545501785001:
    triangular: 5393607.158145172316497304724655
   tetrahedral: 44355.777384073256052620916889
    pentatonic: 4321.000000000000000000000000

J

In J, it's usually more natural to start counting from 0 rather than 1. That shows up subtly in this task, since the specified roots assume counting starts from 1.

Anyways:

   tri=: [!+
   2 tri 1+i.5 6
  3   6  10  15  21  28
 36  45  55  66  78  91
105 120 136 153 171 190
210 231 253 276 300 325
351 378 406 435 465 496
   3 tri 1+i.5 6
   4   10   20   35   56   84
 120  165  220  286  364  455
 560  680  816  969 1140 1330
1540 1771 2024 2300 2600 2925
3276 3654 4060 4495 4960 5456
   4 tri 1+i.5 6
    5    15    35    70   126   210
  330   495   715  1001  1365  1820
 2380  3060  3876  4845  5985  7315
 8855 10626 12650 14950 17550 20475
23751 27405 31465 35960 40920 46376
   12 tri 1+i.10 3
        13         91         455
      1820       6188       18564
     50388     125970      293930
    646646    1352078     2704156
   5200300    9657700    17383860
  30421755   51895935    86493225
 141120525  225792840   354817320
 548354040  834451800  1251677700
1852482996 2707475148  3910797436
5586853480 7898654920 11058116888

And, for the roots:

   r2=: 2 %~ _1 + 2 %: 1 8&p.
   r3=: _1 + 0 3&p. (+ +&(3%:]) -) 2 %: _1r27 0 9&p.
   r4=: 2 %~ _3 + 2 %: 5 + 4 * 2 %: 1 + 24 * ] 
   (r2,r3,r4) 7140
119 34 18.8766
   (r2,r3,r4) 21408696
6543 503.564 149.061
   (r2,r3,r4) 26728085384
231205 5432 893.442
   (r2,r3,r4) 14545501785001
5.39361e6 44356.2 4321

jq

Works with: jq

Also works with gojq and fq, the Go implementations of jq

The main point of interest in the following is probably `figurates/0`, which generates an indefinitely long stream of the $r-simplex numbers if $r >= 2, where $r is the input to the filter. For the sake of illustration, however, `tetrahedrals` and `pentatopics` are defined without reference to `figurates/0`.

Preliminaries

def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;

# Display a stream of items in Z-style, n per line
def neatly(s; $n; $width):
  def p: lpad($width);
  foreach s as $x ({n: 1, s:""};
      if .n >= $n
      then .emit = .s + " " + ($x|p)
      | .s = null
      | .n = 1
      else .emit = null
      | .s = .s + " " + ($x|p)
      | .n += 1
      end;
      select(.emit).emit);

# nCk assuming n >= k
def binomial(n; k): 
  if k > n / 2 then binomial(n; n-k)
  else reduce range(1; k+1) as $i (1; . * (n - $i + 1) / $i)
  end;

def figurate($r; $n): binomial($n + $r -1; $r);

def triangular: binomial(.+1;2);

# r=2
def triangulars: foreach range(0; infinite) as $i (0; . + $i);

# r=3
def tetrahedrals: foreach triangulars as $t (0; . + $t);

# r=4
def pentatopics:  foreach tetrahedrals as $t (0; . + $t);

# input: r
def figurates:
  . as $r
  | if $r == 2 then triangulars
    else foreach ($r - 1 |figurates) as $t (0; . + $t)
    end;
         
# r=12
def twelveSimplexes: 12 | figurates;

###  r-simplex roots

def triangularRoot: ((8*. + 1 | sqrt) -1) /2;

def tetrahedralRoot:
  def term(sign):
     (3 * .) as $y
     | ($y + sign * ( (($y*$y) - (1/27))|sqrt)) | cbrt;
  term(1) + term(-1) -1;

def pentatopicRoot:
   (((5 + 4 * (( 24*. + 1)|sqrt)) | sqrt) - 3) / 2;

def xs: [7140, 21408696, 26728085384, 14545501785001];

def tasks:
  def round($ndec): pow(10;$ndec) as $p | . * $p | round / $p;
  def r: round(4) | lpad(12);
  
  def s(stream): limit(30; neatly(stream; 5; 8));
  "The first 30 triangular numbers are:",    s(triangulars),
  "\nThe first 30 tetrahedral numbers are:", s(tetrahedrals),
  "\nThe first 30 pentatopic numbers are:",  s(pentatopics),
  "\nThe first 30 12-simplex numbers are:",  neatly(limit(30; twelveSimplexes); 5; 12),
  
  "",
  "Approximate r-simplex roots:",
  "\("x "|lpad(15)) triangularRoot tetrahedralRoot  pentatopicRoot",
  (xs[]
   | "\(lpad(15)): \(triangularRoot|r) \(tetrahedralRoot|r) \(pentatopicRoot|r)")

  ;

tasks

Output:

The first 30 triangular numbers are:
        0        1        3        6       10
       15       21       28       36       45
       55       66       78       91      105
      120      136      153      171      190
      210      231      253      276      300
      325      351      378      406      435
      465      496      528      561      595
      630      666      703      741      780
      820      861      903      946      990
     1035     1081     1128     1176     1225
     1275     1326     1378     1431     1485
     1540     1596     1653     1711     1770
     1830     1891     1953     2016     2080
     2145     2211     2278     2346     2415
     2485     2556     2628     2701     2775
     2850     2926     3003     3081     3160
     3240     3321     3403     3486     3570
     3655     3741     3828     3916     4005
     4095     4186     4278     4371     4465
     4560     4656     4753     4851     4950
     5050     5151     5253     5356     5460
     5565     5671     5778     5886     5995
     6105     6216     6328     6441     6555
     6670     6786     6903     7021     7140
     7260     7381     7503     7626     7750
     7875     8001     8128     8256     8385
     8515     8646     8778     8911     9045
     9180     9316     9453     9591     9730
     9870    10011    10153    10296    10440
    10585    10731    10878    11026    11175

The first 30 tetrahedral numbers are:
        0        1        4       10       20
       35       56       84      120      165
      220      286      364      455      560
      680      816      969     1140     1330
     1540     1771     2024     2300     2600
     2925     3276     3654     4060     4495
     4960     5456     5984     6545     7140
     7770     8436     9139     9880    10660
    11480    12341    13244    14190    15180
    16215    17296    18424    19600    20825
    22100    23426    24804    26235    27720
    29260    30856    32509    34220    35990
    37820    39711    41664    43680    45760
    47905    50116    52394    54740    57155
    59640    62196    64824    67525    70300
    73150    76076    79079    82160    85320
    88560    91881    95284    98770   102340
   105995   109736   113564   117480   121485
   125580   129766   134044   138415   142880
   147440   152096   156849   161700   166650
   171700   176851   182104   187460   192920
   198485   204156   209934   215820   221815
   227920   234136   240464   246905   253460
   260130   266916   273819   280840   287980
   295240   302621   310124   317750   325500
   333375   341376   349504   357760   366145
   374660   383306   392084   400995   410040
   419220   428536   437989   447580   457310
   467180   477191   487344   497640   508080
   518665   529396   540274   551300   562475

The first 30 pentatopic numbers are:
        0        1        5       15       35
       70      126      210      330      495
      715     1001     1365     1820     2380
     3060     3876     4845     5985     7315
     8855    10626    12650    14950    17550
    20475    23751    27405    31465    35960
    40920    46376    52360    58905    66045
    73815    82251    91390   101270   111930
   123410   135751   148995   163185   178365
   194580   211876   230300   249900   270725
   292825   316251   341055   367290   395010
   424270   455126   487635   521855   557845
   595665   635376   677040   720720   766480
   814385   864501   916895   971635  1028790
  1088430  1150626  1215450  1282975  1353275
  1426425  1502501  1581580  1663740  1749060
  1837620  1929501  2024785  2123555  2225895
  2331890  2441626  2555190  2672670  2794155
  2919735  3049501  3183545  3321960  3464840
  3612280  3764376  3921225  4082925  4249575
  4421275  4598126  4780230  4967690  5160610
  5359095  5563251  5773185  5989005  6210820
  6438740  6672876  6913340  7160245  7413705
  7673835  7940751  8214570  8495410  8783390
  9078630  9381251  9691375 10009125 10334625
 10668000 11009376 11358880 11716640 12082785
 12457445 12840751 13232835 13633830 14043870
 14463090 14891626 15329615 15777195 16234505
 16701685 17178876 17666220 18163860 18671940
 19190605 19720001 20260275 20811575 21374050

The first 30 12-simplex numbers are:
            0            1           13           91          455
         1820         6188        18564        50388       125970
       293930       646646      1352078      2704156      5200300
      9657700     17383860     30421755     51895935     86493225
    141120525    225792840    354817320    548354040    834451800
   1251677700   1852482996   2707475148   3910797436   5586853480

Approximate r-simplex roots:
             x  triangularRoot tetrahedralRoot  pentatopicRoot
           7140:          119           34      18.8766
       21408696:         6543     503.5612     149.0609
    26728085384:  231205.4056    5431.9999     893.4425
 14545501785001: 5393607.1581   44355.7774         4321

Julia

Translation of: Raku

""" rosettacode.org task Triangular_numbers """


polytopic(r, range) = map(n -> binomial(n + r - 1, r), range)

triangular_root(x) = (sqrt(8x + 1) - 1) / 2

function tetrahedral_root(x)
    return Float64(round((3x + sqrt(9 * x^big"2" - big"1"/27))^(big"1"/3) +
       (3x - sqrt(9 * x^big"2" - big"1"/27))^(big"1"/3) - 1, digits=18))
end

pentatopic_root(x) = (sqrt(5 + 4 * sqrt(24x + 1)) - 3) / 2

function valuelisting(a, N=6)
    c = maximum(length, string.(a)) + 1
    return join([join([lpad(x, c) for x in v]) for v in Iterators.partition(a, N)], "\n")
end

for (r, name) in [[2, "triangular"], [3, "tetrahedral"], [4, "pentatopic"], [12, "12-simplex"]]
    println("\nFirst 30 $name numbers:\n", valuelisting(polytopic(r, 0:29)))
end

for n in [7140, 21408696, 26728085384, 14545501785001]
    println("\nRoots of $n:")
    println("   triangular-root: ", triangular_root(n))
    println("   tetrahedral-root: ", tetrahedral_root(n))
    println("   pentatopic-root: ", pentatopic_root(n))
end

Output:

First 30 triangular numbers:
   0   1   3   6  10  15
  21  28  36  45  55  66
  78  91 105 120 136 153
 171 190 210 231 253 276
 300 325 351 378 406 435

First 30 tetrahedral numbers:
    0    1    4   10   20   35
   56   84  120  165  220  286
  364  455  560  680  816  969
 1140 1330 1540 1771 2024 2300
 2600 2925 3276 3654 4060 4495

First 30 pentatopic numbers:
     0     1     5    15    35    70
   126   210   330   495   715  1001
  1365  1820  2380  3060  3876  4845
  5985  7315  8855 10626 12650 14950
 17550 20475 23751 27405 31465 35960

First 30 12-simplex numbers:
          0          1         13         91        455       1820
       6188      18564      50388     125970     293930     646646
    1352078    2704156    5200300    9657700   17383860   30421755
   51895935   86493225  141120525  225792840  354817320  548354040
  834451800 1251677700 1852482996 2707475148 3910797436 5586853480

Roots of 7140:
   triangular-root: 119.0
   tetrahedral-root: 34.0
   pentatopic-root: 18.876646615928006

Roots of 21408696:
   triangular-root: 6543.0
   tetrahedral-root: 503.5618269746365
   pentatopic-root: 149.06094737526587

Roots of 26728085384:
   triangular-root: 231205.40556525585
   tetrahedral-root: 5432.0
   pentatopic-root: 893.4424567516849

Roots of 14545501785001:
   triangular-root: 5.3936071581451725e6
   tetrahedral-root: 44355.777384073255
   pentatopic-root: 4321.0

Mathematica/Wolfram Language

Translation of: Julia

(* Polytopic number generation function *)
Polytopic[r_, range_] := Binomial[# + r - 1, r] & /@ range

(* Triangular root function *)
TriangularRoot[x_] := (Sqrt[8 x + 1] - 1)/2

(* Tetrahedral root function *)
TetrahedralRoot[x_] := N[((3 x + Sqrt[9 x^2 - 1/27])^(1/3) + 
    (3 x - Sqrt[9 x^2 - 1/27])^(1/3) - 1), 18]

(* Pentatopic root function *)
PentatopicRoot[x_] := (Sqrt[5 + 4 Sqrt[24 x + 1]] - 3)/2

(* Displaying polytopic numbers *)
Do[
  name = Which[
    r == 2, "triangular",
    r == 3, "tetrahedral",
    r == 4, "pentatopic",
    r == 12, "12-simplex"
  ];
  Print["\nFirst 30 ", name, " numbers:\n", Polytopic[r, Range[0, 29]]],
 {r,{2,3,4,12}}
]

(* Displaying roots of specific numbers *)
nums = {7140, 21408696, 26728085384, 14545501785001};
For[i = 1, i <= Length[nums], i++,
  n = nums[[i]];
  Print["\nRoots of ", n, ":"];
  Print["   triangular-root: ", N@TriangularRoot[n]];
  Print["   tetrahedral-root: ", N@TetrahedralRoot[n]];
  Print["   pentatopic-root: ", N@PentatopicRoot[n]]
]

Output:


First 30 triangular numbers:
{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435}

First 30 tetrahedral numbers:
{0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495}

First 30 pentatopic numbers:
{0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960}

First 30 12-simplex numbers:
{0, 1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704156, 5200300, 9657700, 17383860, 30421755, 51895935, 86493225, 141120525, 225792840, 354817320, 548354040, 834451800, 1251677700, 1852482996, 2707475148, 3910797436, 5586853480}

Roots of 7140:
   triangular-root: 119.
   tetrahedral-root: 34.
   pentatopic-root: 18.876646615928006

Roots of 21408696:
   triangular-root: 6543.
   tetrahedral-root: 503.5618269746365
   pentatopic-root: 149.06094737526587

Roots of 26728085384:
   triangular-root: 231205.40556525585
   tetrahedral-root: 5432.
   pentatopic-root: 893.4424567516849

Roots of 14545501785001:
   triangular-root: 5.3936071581451725*^6
   tetrahedral-root: 44355.777384073255
   pentatopic-root: 4321.

Nim

As described in the task presentation, we start the sequences at index 1.

import std/[math, strformat, strutils]


proc printNSimplexNumbers(r, count, width: Positive; title: string) =
  ## Print the first "count" terms of the "r-simplex" sequence
  ## using "width" characters.
  echo title
  for n in 1..count:
    stdout.write align($binom(n + r - 1, r), width)
    stdout.write if n mod 5 == 0: '\n' else: ' '
  echo()

printNSimplexNumbers(2, 30, 3, "First 30 triangular numbers:")
printNSimplexNumbers(3, 30, 4, "First 30 tetrahedral numbers:")
printNSimplexNumbers(4, 30, 5, "First 30 pentatopic numbers:")
printNSimplexNumbers(12, 30, 10, "First 30 12-simplex numbers:")


func triangularRoot(x: float): float =
  ## Return the triangular root of "x".
  (sqrt(8 * x + 1) - 1) * 0.5

func tetrahedralRoot(x: float): float =
  ## Return the tetrahedral root of "x".
  let t1 = 3 * x
  let t2 = sqrt(t1 * t1 - 1 / 27)
  result = cbrt(t1 + t2) + cbrt(t1 - t2) - 1

func pentatopicRoot(x: float): float =
  ## Return the pentatopic root of "x".
  (sqrt(5 + 4 * sqrt(24 * x + 1)) - 3) * 0.5

for n in [int64 7140, 21408696, 26728085384, 14545501785001]:
  echo &"Roots of {n}:"
  for (title, f) in {"triangular: ": triangularRoot,
                     "tetrahedral:": tetrahedralRoot,
                     "pentatopic: ": pentatopicRoot}:
    echo &"  {title} {f(n.float):.6f}"
  echo()

Output:

First 30 triangular numbers:
  1   3   6  10  15
 21  28  36  45  55
 66  78  91 105 120
136 153 171 190 210
231 253 276 300 325
351 378 406 435 465

First 30 tetrahedral numbers:
   1    4   10   20   35
  56   84  120  165  220
 286  364  455  560  680
 816  969 1140 1330 1540
1771 2024 2300 2600 2925
3276 3654 4060 4495 4960

First 30 pentatopic numbers:
    1     5    15    35    70
  126   210   330   495   715
 1001  1365  1820  2380  3060
 3876  4845  5985  7315  8855
10626 12650 14950 17550 20475
23751 27405 31465 35960 40920

First 30 12-simplex numbers:
         1         13         91        455       1820
      6188      18564      50388     125970     293930
    646646    1352078    2704156    5200300    9657700
  17383860   30421755   51895935   86493225  141120525
 225792840  354817320  548354040  834451800 1251677700
1852482996 2707475148 3910797436 5586853480 7898654920

Roots of 7140:
  triangular:  119.000000
  tetrahedral: 34.000000
  pentatopic:  18.876647

Roots of 21408696:
  triangular:  6543.000000
  tetrahedral: 503.561166
  pentatopic:  149.060947

Roots of 26728085384:
  triangular:  231205.405565
  tetrahedral: 5431.999939
  pentatopic:  893.442457

Roots of 14545501785001:
  triangular:  5393607.158145
  tetrahedral: 44355.777377
  pentatopic:  4321.000000

PARI/GP

Translation of: Julia

/* Polytopic number generation function */
polytopic(r, range) = {
    vector(#range, i, binomial(range[i] + r - 1, r))
}

/* Triangular root function */
triangularRoot(x) = {
    (sqrt(8*x + 1) - 1)/2
}

/* Tetrahedral root function */
tetrahedralRoot(x) = {
    N = (3*x + sqrt(9*x^2 - 1/27))^(1/3) + (3*x - sqrt(9*x^2 - 1/27))^(1/3) - 1;
    precision(N, 18)
}

/* Pentatopic root function */
pentatopicRoot(x) = {
    (sqrt(5 + 4*sqrt(24*x + 1)) - 3)/2
}


{
/* Displaying polytopic numbers */
r_sel=[2,3,4,12];
for(i = 1, #r_sel,
    r=r_sel[i];
    casename="place_holder";
    if(r == 2, casename="triangular"; ,
       r == 3, casename="tetrahedral"; ,
       r == 4, casename="pentatopic"; ,
       r == 12, casename="12-simplex";
     );
    printf("\nFirst 30 %s numbers:\n %s\n" , Str(casename) , Str(polytopic(r, [0..29])) )
);

/* Displaying roots of specific numbers */
nums = [7140, 21408696, 26728085384, 14545501785001];
for(i = 1, #nums,
    n = nums[i];
    printf("\nRoots of %s:\n", Str(n));
    printf("   triangular-root: %s\n", Str(triangularRoot(n)));
    printf("   tetrahedral-root: %s\n", Str(tetrahedralRoot(n)));
    printf("   pentatopic-root: %s\n",  Str(pentatopicRoot(n)))
);
}

Output:


First 30 triangular numbers:
 [0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435]

First 30 tetrahedral numbers:
 [0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495]

First 30 pentatopic numbers:
 [0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960]

First 30 12-simplex numbers:
 [0, 1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704156, 5200300, 9657700, 17383860, 30421755, 51895935, 86493225, 141120525, 225792840, 354817320, 548354040, 834451800, 1251677700, 1852482996, 2707475148, 3910797436, 5586853480]

Roots of 7140:
   triangular-root: 119.00000000000000000000000000000000000
   tetrahedral-root: 34.00000000000000000
   pentatopic-root: 18.876646615928006607901782826667566229

Roots of 21408696:
   triangular-root: 6543.0000000000000000000000000000000000
   tetrahedral-root: 503.5618269746365141
   pentatopic-root: 149.06094737526586748438757488471336807

Roots of 26728085384:
   triangular-root: 231205.40556525583695729103196069412230
   tetrahedral-root: 5432.000000000000000
   pentatopic-root: 893.44245675168486988846621152924537039

Roots of 14545501785001:
   triangular-root: 5393607.1581451723164973047246554846080
   tetrahedral-root: 44355.77738407325605
   pentatopic-root: 4321.0000000000000000000000000000000000

Pascal

Using only extended isn't that precise for tetrahedral roots.
sqrt(sqr(3x)+1/27) is nearly 3x for bigger x values.

program XangularNumbers;
const 
  MAXIDX = 29;
  MAXLINECNT = 13;
  cNames : array[0..4] of string =
     ('','','triangular','tetrahedral','pentatopic');
  cCheckRootValues :array[0..3] of Uint64 = 
       (7140,21408696,26728085384,14545501785001)   ;
type
  tOneLine  = array[0..MAXIDX+2] of Uint64;
  tpOneLine = ^tOneLine;
  tSimplexs  = array[0..MAXLINECNT-1] of tOneLine;  

procedure OutLine(var S:tSimplexs;idx: NativeInt);
const
  cColCnt = 6;cColWidth = 80 DIV cColCnt;
var
  i,colcnt : NativeInt;
begin
  if idx > High(cNames) then
    writeln('First ',MAXIDX+1,' ',idx,'-simplex numbers')  
  else
    writeln('First ',MAXIDX+1,' ',cNames[idx],' numbers');
  colcnt := cColCnt;
  For i := 0 to MAXIDX do
  begin
    write(S[idx,i]:cColWidth);
    dec(colCnt);
    if ColCnt = 0 then
    Begin
      writeln;
      ColCnt := cColCnt;
    end;
  end; 
  if ColCnt <  cColCnt then
    writeln;
  writeln;   
end;  

procedure CalcNextLine(var S:tSimplexs;idx: NativeInt);  
var
  s1,s2: Uint64;  
  i : NativeInt;
begin
  s1 := S[idx,0];  
  S[idx+1,0] := s1;
  For i := 1 to MAXIDX do
  begin
    s2:= S[idx,i];
    S[idx+1,i] := s1+s2;
    inc(s1,s2);
  end;  
end;

procedure InitSimplexs(var S:tSimplexs);
var
  i: NativeInt;
begin
  fillChar(S,Sizeof(S),#0);
  For i := 1 to MAXIDX do
    S[0,i] := 1;
  For i := 0 to MAXLINECNT-2 do    
    CalcNextLine(S,i);    
end;

function TriangularRoot(n: Uint64): extended;
begin
  if n < High(Uint64) DIV 8 then
    TriangularRoot := (sqrt(8*n+1)-1) / 2
  else
    TriangularRoot := (sqrt(8)*sqrt(n)-1)/2;
end;  

function tetrahedralRoot(n: Uint64): extended;
const
  cRec27 = 1/sqrt(27);
var
  x,y : extended;
begin
  y := 3.0*n;  
  x := sqrt((y-cRec27)*(y+cRec27));//sqrt(sqr(3*n)-1/27)
  if x < y then
    tetrahedralRoot := exp(ln(y+x)/3.0)+exp(ln(y-x)/3.0)-1.0
  else
    //( 6*n)^(1/3)-1
    tetrahedralRoot :=exp(ln(6)/3.0)*exp(ln(n)/3.0)-1.0; //6^(1/3)* n^(1/3)-1  
end;

function PentatopicRoot(n: Uint64): extended;
begin
  PentatopicRoot := (sqrt(5 + 4 * sqrt(24*n + 1)) - 3) / 2;
end; 

var
  Simplexs  : tSimplexs;
  n : Uint64;
  i : NativeInt;
Begin
  InitSimplexs(Simplexs);
  OutLine(Simplexs,2);
  OutLine(Simplexs,3);
  OutLine(Simplexs,4);
  OutLine(Simplexs,12);
  For i := 0 to High(cCheckRootValues) do
  begin
    n := cCheckRootValues[i];
    writeln('Roots of ',n,':');
    writeln('triangular -root : ',TriangularRoot(n):20:12);
    writeln('tetrahedral-root : ',tetrahedralRoot(n):20:12);
    writeln('pentatopic -root : ',PentatopicRoot(n):20:12);
    writeln;
  end;
end.

Output:

First 30 triangular numbers
            0            1            3            6           10           15
           21           28           36           45           55           66
           78           91          105          120          136          153
          171          190          210          231          253          276
          300          325          351          378          406          435

First 30 tetrahedral numbers
            0            1            4           10           20           35
           56           84          120          165          220          286
          364          455          560          680          816          969
         1140         1330         1540         1771         2024         2300
         2600         2925         3276         3654         4060         4495

First 30 pentatopic numbers
            0            1            5           15           35           70
          126          210          330          495          715         1001
         1365         1820         2380         3060         3876         4845
         5985         7315         8855        10626        12650        14950
        17550        20475        23751        27405        31465        35960

First 30 12-simplex numbers
            0            1           13           91          455         1820
         6188        18564        50388       125970       293930       646646
      1352078      2704156      5200300      9657700     17383860     30421755
     51895935     86493225    141120525    225792840    354817320    548354040
    834451800   1251677700   1852482996   2707475148   3910797436   5586853480

Roots of 7140:
triangular -root :     119.000000000000
tetrahedral-root :      34.000000000003
pentatopic -root :      18.876646615928

Roots of 21408696:
triangular -root :    6543.000000000000
tetrahedral-root :     503.561826261328
pentatopic -root :     149.060947375266

Roots of 26728085384:
triangular -root :  231205.405565255837
tetrahedral-root :    5431.999938646542 <<==
pentatopic -root :     893.442456751685

Roots of 14545501785001:
triangular -root : 5393607.158145172316
tetrahedral-root :   44355.777376558433
pentatopic -root :    4321.000000000000

Perl

Translation of: Raku

use v5.36;
use experimental <builtin for_list>;
use Math::AnyNum <binomial cbrt max round>;

sub table { my $t = 6 * (my $c = 1 + length max @_); ( sprintf( ('%'.$c.'d')x@_, @_) ) =~ s/.{1,$t}\K/\n/gr }

sub triangular_root ($x) {
    round( (sqrt(8 * $x + 1) - 1) / 2, -3);
}

sub tetrahedral_root ($x) {
    round(
      cbrt(3 * $x + sqrt 9 * $x**2 - 1/27) +
      cbrt(3 * $x - sqrt 9 * $x**2 - 1/27) - 1,
    -3)
}

sub pentatopic_root ($x) {
    round( (sqrt(5 + 4 * sqrt 24 * $x + 1) - 3) / 2, -3)
}

sub polytopic ($r, @range) { map { binomial $_ + $r - 1, $r } @range }

for my($r,$label) (2, 'triangular', 3, 'tetrahedral', 4, 'pentatopic', 12, '12-simplex') {
    say "First 30 $label numbers:\n" . table polytopic $r, 0..29
}

for (7140, 21408696, 26728085384, 14545501785001) {
    printf "Roots of $_:
   triangular-root: %s
  tetrahedral-root: %s
   pentatopic-root: %s\n\n",
            triangular_root($_), tetrahedral_root($_), pentatopic_root($_);
}

Output:

First 30 triangular numbers:
   0   1   3   6  10  15
  21  28  36  45  55  66
  78  91 105 120 136 153
 171 190 210 231 253 276
 300 325 351 378 406 435

First 30 tetrahedral numbers:
    0    1    4   10   20   35
   56   84  120  165  220  286
  364  455  560  680  816  969
 1140 1330 1540 1771 2024 2300
 2600 2925 3276 3654 4060 4495

First 30 pentatopic numbers:
     0     1     5    15    35    70
   126   210   330   495   715  1001
  1365  1820  2380  3060  3876  4845
  5985  7315  8855 10626 12650 14950
 17550 20475 23751 27405 31465 35960

First 30 12-simplex numbers:
          0          1         13         91        455       1820
       6188      18564      50388     125970     293930     646646
    1352078    2704156    5200300    9657700   17383860   30421755
   51895935   86493225  141120525  225792840  354817320  548354040
  834451800 1251677700 1852482996 2707475148 3910797436 5586853480

Roots of 7140:
   triangular-root: 119
  tetrahedral-root: 34
   pentatopic-root: 18.877

Roots of 21408696:
   triangular-root: 6543
  tetrahedral-root: 503.561
   pentatopic-root: 149.061

Roots of 26728085384:
   triangular-root: 231205.406
  tetrahedral-root: 5432
   pentatopic-root: 893.442

Roots of 14545501785001:
   triangular-root: 5393607.158
  tetrahedral-root: 44355.777
   pentatopic-root: 4321

Phix

with javascript_semantics
sequence t = repeat(0,30)
for n=2 to 30 do t[n] = t[n-1] + n-1 end for
printf(1,"The first 30 triangular numbers are:\n%s\n",{join_by(t,1,6,fmt:="%3d")})
for n=2 to 30 do t[n] = t[n] + t[n-1] end for
printf(1,"The first 30 tetrahedral numbers are:\n%s\n",{join_by(t,1,6,fmt:="%4d")})
for n=2 to 30 do t[n] = t[n] + t[n-1] end for
printf(1,"The first 30 pentatopic numbers are:\n%s\n",{join_by(t,1,6,fmt:="%5d")})
for r=5 to 12 do
    for n=2 to 30 do t[n] = t[n] + t[n-1] end for
end for
printf(1,"The first 30 12-simplex numbers are:\n%s\n",{join_by(t,1,6,fmt:="%10d")})

for x in {7140, 21408696, 26728085384, 14545501785001} do
    printf(1,"\nRoots of %d:\n",x)
    atom root = (sqrt(x*8 + 1)-1)/2
    printf(1,"%14s: %f\n", {"triangular", root})

    atom temp = sqrt(x*x*9 - 1/27)
    root = power(x*3 + temp,1/3) + power(x*3 - temp,1/3) - 1
    printf(1,"%14s: %f\n", {"tetrahedral", root})

    root = (sqrt(sqrt(x*24 + 1)*4 + 5) - 3) / 2
    printf(1,"%14s: %f\n", {"pentatopic", root})
end for

Output:

The first 30 triangular numbers are:
  0     1     3     6    10    15
 21    28    36    45    55    66
 78    91   105   120   136   153
171   190   210   231   253   276
300   325   351   378   406   435

The first 30 tetrahedral numbers are:
   0      1      4     10     20     35
  56     84    120    165    220    286
 364    455    560    680    816    969
1140   1330   1540   1771   2024   2300
2600   2925   3276   3654   4060   4495

The first 30 pentatopic numbers are:
    0       1       5      15      35      70
  126     210     330     495     715    1001
 1365    1820    2380    3060    3876    4845
 5985    7315    8855   10626   12650   14950
17550   20475   23751   27405   31465   35960

The first 30 12-simplex numbers are:
         0            1           13           91          455         1820
      6188        18564        50388       125970       293930       646646
   1352078      2704156      5200300      9657700     17383860     30421755
  51895935     86493225    141120525    225792840    354817320    548354040
 834451800   1251677700   1852482996   2707475148   3910797436   5586853480

Roots of 7140:
    triangular: 119.000000
   tetrahedral: 34.000000
    pentatopic: 18.876647

Roots of 21408696:
    triangular: 6543.000000
   tetrahedral: 503.561166
    pentatopic: 149.060947

Roots of 26728085384:
    triangular: 231205.405565
   tetrahedral: 5431.999939
    pentatopic: 893.442457

Roots of 14545501785001:
    triangular: 5393607.158145
   tetrahedral: 44355.777377
    pentatopic: 4321.000000

Raku

use Math::Root:ver<0.0.4>;

sub binomial { [×] ($^n … 0) Z/ 1 .. $^p }

sub polytopic (Int $r, @range) { @range.map: { binomial $_ + $r - 1, $r } }

sub display (@values) {
    my $c = @values.max.chars;
    @values.batch(6)».fmt("%{$c}d").join: "\n";
}

for 2, 'triangular', 3, 'tetrahedral', 4, 'pentatopic', 12, '12-simplex'
  -> $r, $name { say "\nFirst 30 $name numbers:\n" ~ display polytopic $r, ^30 }

say '';

my \ε = FatRat.new: 1, 10**24;

for 7140, 21408696, 26728085384, 14545501785001 {
  say qq:to/R/;
  Roots of $_:
    triangular-root: {.&triangular-root.round:  ε}
   tetrahedral-root: {.&tetrahedral-root.round: ε}
    pentatopic-root: {.&pentatopic-root.round:  ε}
  R
}

Output:

First 30 triangular numbers:
  0   1   3   6  10  15
 21  28  36  45  55  66
 78  91 105 120 136 153
171 190 210 231 253 276
300 325 351 378 406 435

First 30 tetrahedral numbers:
   0    1    4   10   20   35
  56   84  120  165  220  286
 364  455  560  680  816  969
1140 1330 1540 1771 2024 2300
2600 2925 3276 3654 4060 4495

First 30 pentatopic numbers:
    0     1     5    15    35    70
  126   210   330   495   715  1001
 1365  1820  2380  3060  3876  4845
 5985  7315  8855 10626 12650 14950
17550 20475 23751 27405 31465 35960

First 30 12-simplex numbers:
         0          1         13         91        455       1820
      6188      18564      50388     125970     293930     646646
   1352078    2704156    5200300    9657700   17383860   30421755
  51895935   86493225  141120525  225792840  354817320  548354040
 834451800 1251677700 1852482996 2707475148 3910797436 5586853480

Roots of 7140:
  triangular-root: 119
 tetrahedral-root: 34
  pentatopic-root: 18.876646615928006607901783

Roots of 21408696:
  triangular-root: 6543
 tetrahedral-root: 503.56182697463651404819613
  pentatopic-root: 149.060947375265867484387575

Roots of 26728085384:
  triangular-root: 231205.405565255836957291031961
 tetrahedral-root: 5432
  pentatopic-root: 893.442456751684869888466212

Roots of 14545501785001:
  triangular-root: 5393607.158145172316497304724655
 tetrahedral-root: 44355.777384073256052620916903
  pentatopic-root: 4321

RPL

≪ DUP ROT + 1 - SWAP COMB
≫ 'SMPLX' STO      

≪ 8 * 1 + √ 1 - 2 /
≫ 'TROOT' STO        

≪ DUP SQ 9 * 27 INV - √ SWAP 3 * 
   DUP2 + 3 INV ^ SWAP ROT - 3 INV ^ + 1 -
≫ 'TeROOT' STO        

≪ 24 * 1 + √ 4 * 5 + √ 3 - 2 /
≫ 'PROOT' STO 

≪ {} 1 30 FOR n n 3 PICK SMPLX + NEXT 
≫ 'SPX30' STO

≪ {7140 21408696 26728085384 14545501785001}
   1 4 FOR n 
      DUP n GET {} 
      OVER TROOT + OVER TeROOT + SWAP PROOT + SWAP 
   NEXT DROP
≫ 'ROOTS' STO

Input:

2 SPX30
3 SPX30
4 SPX30
12 SPX30
ROOTS

Output:

8: { 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 231 253 276 300 325 351 378 406 435 465 }
7: { 1 4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1140 1330 1540 1771 2024 2300 2600 2925 3276 3654 4060 4495 4960 }
6: { 1 5 15 35 70 126 210 330 495 715 1001 1365 1820 2380 3060 3876 4845 5985 7315 8855 10626 12650 14950 17550 20475 23751 27405 31465 35960 40920 }
5: { 1 13 91 455 1820 6188 18564 50388 125970 293930 646646 1352078 2704156 5200300 9657700 17383860 30421755 51895935 86493225 141120525 225792840 354817320 548354040 834451800 1251677700 1852482996 2707475148 3910797436 5586853480 7898654920 }
4: { 119 34.0000000018 18.8766466159 }
3: { 6543 503.561166335 149.060947375 }
2: { 231205.405565 5431.99993865 893.442456752 }
1: { 5393607.15814 44355.7773766 4321 }

Sidef

func pentatopic_root(x) {
    (sqrt(5 + 4*sqrt(24*x + 1)) - 3)/2
}

func polytopic (r, range) {
    range.map {|n| binomial(n + r - 1, r) }
}

[
    2, 'triangular', 3, 'tetrahedral', 4, 'pentatopic', 12, '12-simplex'
].slices(2).each_2d {|r,label|
    say "\nFirst 30 #{label} numbers:"
    polytopic(r, ^30).slices(6).each{.join(' ').say}
}

for n in (7140, 21408696, 26728085384, 14545501785001) {
    printf ("\nRoots of #{n}:
   triangular-root: %s
  tetrahedral-root: %s
   pentatopic-root: %s\n",
        polygonal_root(n,3), pyramidal_root(n,3), pentatopic_root(n))
}

Output:

First 30 triangular numbers:
0 1 3 6 10 15
21 28 36 45 55 66
78 91 105 120 136 153
171 190 210 231 253 276
300 325 351 378 406 435

First 30 tetrahedral numbers:
0 1 4 10 20 35
56 84 120 165 220 286
364 455 560 680 816 969
1140 1330 1540 1771 2024 2300
2600 2925 3276 3654 4060 4495

First 30 pentatopic numbers:
0 1 5 15 35 70
126 210 330 495 715 1001
1365 1820 2380 3060 3876 4845
5985 7315 8855 10626 12650 14950
17550 20475 23751 27405 31465 35960

First 30 12-simplex numbers:
0 1 13 91 455 1820
6188 18564 50388 125970 293930 646646
1352078 2704156 5200300 9657700 17383860 30421755
51895935 86493225 141120525 225792840 354817320 548354040
834451800 1251677700 1852482996 2707475148 3910797436 5586853480

Roots of 7140:
   triangular-root: 119
  tetrahedral-root: 34
   pentatopic-root: 18.8766466159280066079017828266675662291603339398

Roots of 21408696:
   triangular-root: 6543
  tetrahedral-root: 503.56182697463651404819613028417773405650502954
   pentatopic-root: 149.060947375265867484387574884713368069543117436

Roots of 26728085384:
   triangular-root: 231205.405565255836957291031960694122304324644392
  tetrahedral-root: 5432
   pentatopic-root: 893.442456751684869888466211529245370387840101701

Roots of 14545501785001:
   triangular-root: 5393607.1581451723164973047246554846079685622181
  tetrahedral-root: 44355.7773840732560526209168894228874431786835756
   pentatopic-root: 4321

Wren

Library: Wren-fmt

Library: Wren-big

import "./fmt" for Fmt
import "./big" for BigRat

var t = List.filled(30, 0)
for (n in 1..29) t[n] = t[n-1] + n
System.print("The first 30 triangular numbers are:")
Fmt.tprint("$3d", t, 6)

for (n in 1..29) t[n] = t[n] + t[n-1]
System.print("\nThe first 30 tetrahedral numbers are:")
Fmt.tprint("$4d", t, 6)

for (n in 1..29) t[n] = t[n] + t[n-1]
System.print("\nThe first 30 pentatopic numbers are:")
Fmt.tprint("$5d", t, 6)

for (r in 5..12) {
    for (n in 1..29) t[n] = t[n] + t[n-1]
}
System.print("\nThe first 30 12-simplex numbers are:")
Fmt.tprint("$10d", t, 6)

var xs = [7140, 21408696, 26728085384, 14545501785001]
var digs = 16
for (x in xs) {
    var bx = BigRat.new(x)
    System.print("\nRoots of %(x):")
    var root = ((bx*8 + 1).sqrt(digs) - 1)/2
    Fmt.print("$14s: $s", "triangular", root.toDecimal(digs-5))

    var temp = (bx*bx*9 - BigRat.new(1, 27)).sqrt(digs)
    root = (bx*3 + temp).cbrt(digs) + (bx*3 - temp).cbrt(digs) - 1
    Fmt.print("$14s: $s", "tetrahedral", root.toDecimal(digs-5))

    root = (((bx*24 + 1).sqrt(digs)*4 + 5).sqrt(digs) - 3) / 2
    Fmt.print("$14s: $s", "pentatopic", root.toDecimal(digs-5))
}

Output:

The first 30 triangular numbers are:
  0   1   3   6  10  15 
 21  28  36  45  55  66 
 78  91 105 120 136 153 
171 190 210 231 253 276 
300 325 351 378 406 435 

The first 30 tetrahedral numbers are:
   0    1    4   10   20   35 
  56   84  120  165  220  286 
 364  455  560  680  816  969 
1140 1330 1540 1771 2024 2300 
2600 2925 3276 3654 4060 4495 

The first 30 pentatopic numbers are:
    0     1     5    15    35    70 
  126   210   330   495   715  1001 
 1365  1820  2380  3060  3876  4845 
 5985  7315  8855 10626 12650 14950 
17550 20475 23751 27405 31465 35960 

The first 30 12-simplex numbers are:
         0          1         13         91        455       1820 
      6188      18564      50388     125970     293930     646646 
   1352078    2704156    5200300    9657700   17383860   30421755 
  51895935   86493225  141120525  225792840  354817320  548354040 
 834451800 1251677700 1852482996 2707475148 3910797436 5586853480 

Roots of 7140:
    triangular: 119
   tetrahedral: 34.00000000000
    pentatopic: 18.87664661593

Roots of 21408696:
    triangular: 6543
   tetrahedral: 503.56182697464
    pentatopic: 149.06094737527

Roots of 26728085384:
    triangular: 231205.40556525584
   tetrahedral: 5432.00000000000
    pentatopic: 893.44245675168

Roots of 14545501785001:
    triangular: 5393607.15814517232
   tetrahedral: 44355.77738407326
    pentatopic: 4321

XPL0

Some "interesting" loss of precision in the Pow function....

real T(13, 30);

proc ShowRoots(X);
real X, SR, CR1, CR2;
[Format(1, 0);
Text(0, "Roots of ");  RlOut(0, X);  CrLf(0);
Format(7, 13);
Text(0, " triangular: ");
RlOut(0, (sqrt(8.*X + 1.) - 1.) / 2.);
Text(0, "^m^jtetrahedral: ");
SR:= sqrt(9.*X*X - 1./27.);
CR1:= Pow(3.*X + SR, 1./3.);
CR2:= Pow(3.*X - SR, 1./3.);
RlOut(0, CR1 + CR2 -1.);
Text(0, "^m^j pentatopic: ");
RlOut(0, (sqrt(5. + 4.*sqrt(24.*X + 1.)) - 3.) / 2.);
CrLf(0);  CrLf(0);
];

proc Print(Str, Places, R);
int  Str, Places, R, N;
[Text(0, Str);  CrLf(0);
Format(Places, 0);
for N:= 0 to 29 do
    [RlOut(0, T(R,N));
    if rem(N/6) = 5 then CrLf(0);
    ];
CrLf(0);
];

int  R, N;
[for N:= 0 to 29 do
    T(1,N):= float(N);
for R:= 2 to 12 do
    [T(R,0):= 0.;
    for N:= 1 to 29 do
        T(R,N):= T(R,N-1) + T(R-1,N);
    ];
Print("The first 30 triangular numbers are:", 4, 2);
Print("The first 30 tetrahedral numbers are:", 5, 3);
Print("The first 30 pentatopic numbers are:", 6, 4);
Print("The first 30 12-simplex numbers are:", 11, 12);
ShowRoots(7140.);
ShowRoots(21408696.);
ShowRoots(26728085384.);
ShowRoots(14_545_501_785_001.);
]

Output:

The first 30 triangular numbers are:
   0   1   3   6  10  15
  21  28  36  45  55  66
  78  91 105 120 136 153
 171 190 210 231 253 276
 300 325 351 378 406 435

The first 30 tetrahedral numbers are:
    0    1    4   10   20   35
   56   84  120  165  220  286
  364  455  560  680  816  969
 1140 1330 1540 1771 2024 2300
 2600 2925 3276 3654 4060 4495

The first 30 pentatopic numbers are:
     0     1     5    15    35    70
   126   210   330   495   715  1001
  1365  1820  2380  3060  3876  4845
  5985  7315  8855 10626 12650 14950
 17550 20475 23751 27405 31465 35960

The first 30 12-simplex numbers are:
          0          1         13         91        455       1820
       6188      18564      50388     125970     293930     646646
    1352078    2704156    5200300    9657700   17383860   30421755
   51895935   86493225  141120525  225792840  354817320  548354040
  834451800 1251677700 1852482996 2707475148 3910797436 5586853480

Roots of 7140
 triangular:     119.0000000000000
tetrahedral:      34.0000000017903
 pentatopic:      18.8766466159280

Roots of 21408696
 triangular:    6543.0000000000000
tetrahedral:     503.5611663345480
 pentatopic:     149.0609473752660

Roots of 26728085384
 triangular:  231205.4055652560000
tetrahedral:    5431.9999386465400
 pentatopic:     893.4424567516850

Roots of 14545501785001
 triangular: 5393607.1581451700000
tetrahedral:   44355.7773765584000
 pentatopic:    4321.0000000000000