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.
Triangular numbers may be calculated by the following explicit formulas:
where 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 may be found using:
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.
Or, may be calculated directly: (Binomial "n plus two choose three".)
One may find the real tetrahedral root using the formula:
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, or (Binomial "n plus three choose four".)
Pentatopic real roots may be found using:
In general, these all belong to the class figurate numbers as they are
based on r dimensional figures. Sometimes they are referred to as r-simplex
numbers. In geometry a simplex is the simplest r-dimensional
object possible.
You may easily extend to an arbitrary dimension r using binomials. Each term n in dimension r is
There is no known general formula to find roots of higher r-simplex numbers.
- 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:
- 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
Raku
use Math::Root;
my \ε = FatRat.new: 1, 10**24;
sub binomial { [×] ($^n … 0) Z/ 1 .. $^p }
sub polytopic (Int $r, @range) { @range.map: { binomial $_ + $r - 1, $r } }
sub triangular-root ($x) { (((8 × $x + 1).&root - 1) / 2).round: ε }
sub tetrahedral-root ($x) {
((3 × $x + (9 × $x² - 1/27).&root).&root(3) +
(3 × $x - (9 × $x² - 1/27).&root).&root(3) - 1).round: ε
}
sub pentatopic-root ($x) { (((5 + 4 × (24 × $x + 1).&root).&root - 3) / 2).round: ε }
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, ^Inf )[^30]
}
say '';
for 21408696, 26728085384, 14545501785001 {
say qq:to/R/;
Roots of $_:
triangular-root: {.&triangular-root}
tetrahedral-root: {.&tetrahedral-root}
pentatopic-root: {.&pentatopic-root}
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 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