Triangular numbers: Difference between revisions
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pentatopic-root: 4321.0
</pre>
=={{header|Mathematica}}/{{header|Wolfram Language}}==
{{trans|Julia}}
<syntaxhighlight lang="Mathematica">
(* 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]]
]
</syntaxhighlight>
{{out}}
<pre>
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.
</pre>
=={{header|Nim}}==
Line 1,016 ⟶ 1,092:
tetrahedral: 44355.777377
pentatopic: 4321.000000
</pre>
=={{header|PARI/GP}}==
{{trans|Julia}}
<syntaxhighlight lang="PARI/GP">
/* 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)))
);
}
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
Line 1,551 ⟶ 1,714:
{{libheader|Wren-fmt}}
{{libheader|Wren-big}}
<syntaxhighlight lang="
import "./big" for BigRat
| |||