Math::N...rahedral(3)

💖 Math::NumSeq::Tetrahedral 💖

okay bestie so we're in the User Contributed Perl Documentation section, and today's main character is Math::NumSeq::Tetrahedral — yes that's the whole name, say it with your chest 💅

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✨ NAME ✨

Math::NumSeq::Tetrahedral — tetrahedral numbers i*(i+1)*(i+2)/6

like. that's the vibe. lil pyramid-shaped numbers. iconic behavior fr 🎀

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🌸 SYNOPSIS 🌸

use Math::NumSeq::Tetrahedral;
my $seq = Math::NumSeq::Tetrahedral->new;
my ($i, $value) = $seq->next;

that's it. that's the whole entrance. you make the object, you call next, the numbers come to you. no cap, it's giving effortless 🌈

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🦋 DESCRIPTION 🦋

So. The tetrahedral numbers, which are literally i*(i+1)*(i+2)/6.

0, 1, 4, 10, 20, 35, 56, 84, 120, ...

look at them. stacking up like lil cannonballs in a triangle-based pyramid. obsessed. they start at 0 and they just keep going, because that's what sequences DO 💜

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💫 FUNCTIONS 💫

okay so for the shared behavior that every sequence class has, go peek at "FUNCTIONS" in Math::NumSeq — that's the common lore, iykyk ✨

• $seq = Math::NumSeq::Tetrahedral->new () — makes you a brand new sequence object and hands it over. that's your girl. take care of her 🫶

🔥 Random Access 🔥

• $value = $seq->ith($i) — returns $i*($i+1)*($i+2)/6. you ask for index $i, you get the tetrahedral number sitting at that spot. direct. no games 💖
• $bool = $seq->pred($value) — returns true if $value has the form i*(i+1)*(i+2)/6 for some positive integer i. so like. is this number part of the clique or not. vibe check 😌
• $i = $seq->value_to_i_floor($value) — gives you back the index i of $value, or if $value isn't one of them, the index of the next tetrahedral number below $value. it rounds down, bestie. always down ⬇️
• $i = $seq->value_to_i_estimate($value) — hands you an estimate of the i that corresponds to $value.

lowkey important: in the current code this $i gives the tetrahedral above or below $value, so it's out by no more than 1. so it's a guess, but a really really good one. don't be dramatic about it 🍓

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🧚 FORMULAS 🧚

⭐ Value to i Estimate ⭐

i*(i+1)*(i+2) always fall in between cubes, so

T(i) = i*(i+1)*(i+2)/6
     = (i^3 + 3*i^2 + 2*i)/6

i^3 < 6*T(i) < (i+1)^3

see how it's all cozy between two cubes?? sandwiched. sandwiched like a lil oreo 😭

for value_to_i_estimate() it's enough to apply a cube root,

i_estimate = floor(cbrt(6*value))

cube root and floor it, done, slay 💅

💜 Value to i Floor 💜

okay so for value_to_i_floor() the cube root can be 1 too big when the given value is in between successive T() tetrahedrals. for example if value=57 then floor(cbrt(6*57))=6 is correct, but value=58 gives floor(cbrt(6*58))=7 which is 1 too big. rude of it honestly 🙈

i = floor(cbrt(6*value))
if i*(i+1)*(i+2) <= 6*value
then i_floor = i
else i_floor = i-1    # cbrt was 1 too big

so we double check and if the cube root overstepped, we knock it back down by one. correction arc complete ✨

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👀 SEE ALSO 👀

Math::NumSeq, Math::NumSeq::Cubes

go read those too. sibling energy fr 🫶

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🏠 HOME PAGE 🏠

<http://user42.tuxfamily.org/math-numseq/index.html>

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👑 LICENSE 👑

Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde

Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.

💀 no warranty bestie. it's free software, it's given to you as-is, and if it breaks that's between you and the GPL. read the license, it's the whole point. periodt.

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and that's the whole page!! this one ships as perl v5.42.0, dated 2025-07-25, under the manual name Math::N...rahedral(3) 💖 mwah, go count some pyramids 🎀✨