NAME
Math::NumSeq::Tetrahedral  tetrahedral numbers i*(i+1)*(i+2)/6
SYNOPSIS
use Math::NumSeq::Tetrahedral;
my $seq = Math::NumSeq::Tetrahedral>new;
my ($i, $value) = $seq>next;
DESCRIPTION
The tetrahedral numbers, i*(i+1)*(i+2)/6.
0, 1, 4, 10, 20, 35, 56, 84, 120, ...
FUNCTIONS
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
Random Access
$value = $seq>ith($i)

Return
$i*($i+1)*($i+2)/6
. $bool = $seq>pred($value)

Return true if
$value
has the form i*(i+1)*(i+2)/6 for some positive integer i. $i = $seq>value_to_i_floor($value)

Return the index i of
$value
or of the next tetrahedral number below$value
. $i = $seq>value_to_i_estimate($value)

Return an estimate of the i corresponding to
$value
.In the current code this
$i
gives the tetrahedral above or below$value
, so is out by no more than 1.
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
For value_to_i_estimate()
it's enough to apply a cube root,
i_estimate = floor(cbrt(6*value))
Value to i Floor
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 floor(cbrt(6*57))=6 is correct, but value=58 floor(cbrt(6*58))=7 is 1 too big.
i = floor(cbrt(6*value))
if i*(i+1)*(i+2) <= 6*value
then i_floor = i
else i_floor = i1 # cbrt was 1 too big
SEE ALSO
Math::NumSeq, Math::NumSeq::Cubes
HOME PAGE
http://user42.tuxfamily.org/mathnumseq/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
MathNumSeq 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.
MathNumSeq 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 MathNumSeq. If not, see <http://www.gnu.org/licenses/>.