'\"
'\" Generated from file 'figurate\&.man' by tcllib/doctools with format 'nroff'
'\"
.TH "math::figurate" 3tcl 1\&.0 tcllib "Tcl Math Library"
.\" The -*- nroff -*- definitions below are for supplemental macros used
.\" in Tcl/Tk manual entries.
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.BS
.SH NAME
math::figurate \- Evaluate figurate numbers
.SH SYNOPSIS
package require \fBTcl  8\&.6\fR
.sp
package require \fBmath::figurate  1\&.0\fR
.sp
\fB::math::figurate::sum_sequence\fR \fIn\fR
.sp
\fB::math::figurate::sum_squares\fR \fIn\fR
.sp
\fB::math::figurate::sum_cubes\fR \fIn\fR
.sp
\fB::math::figurate::sum_4th_power\fR \fIn\fR
.sp
\fB::math::figurate::sum_5th_power\fR \fIn\fR
.sp
\fB::math::figurate::sum_6th_power\fR \fIn\fR
.sp
\fB::math::figurate::sum_7th_power\fR \fIn\fR
.sp
\fB::math::figurate::sum_8th_power\fR \fIn\fR
.sp
\fB::math::figurate::sum_9th_power\fR \fIn\fR
.sp
\fB::math::figurate::sum_10th_power\fR \fIn\fR
.sp
\fB::math::figurate::sum_sequence_odd\fR \fIn\fR
.sp
\fB::math::figurate::sum_squares_odd\fR \fIn\fR
.sp
\fB::math::figurate::sum_cubes_odd\fR \fIn\fR
.sp
\fB::math::figurate::sum_4th_power_odd\fR \fIn\fR
.sp
\fB::math::figurate::sum_5th_power_odd\fR \fIn\fR
.sp
\fB::math::figurate::sum_6th_power_odd\fR \fIn\fR
.sp
\fB::math::figurate::sum_7th_power_odd\fR \fIn\fR
.sp
\fB::math::figurate::sum_8th_power_odd\fR \fIn\fR
.sp
\fB::math::figurate::sum_9th_power_odd\fR \fIn\fR
.sp
\fB::math::figurate::sum_10th_power_odd\fR \fIn\fR
.sp
\fB::math::figurate::oblong\fR \fIn\fR
.sp
\fB::math::figurate::pronic\fR \fIn\fR
.sp
\fB::math::figurate::triangular\fR \fIn\fR
.sp
\fB::math::figurate::square\fR \fIn\fR
.sp
\fB::math::figurate::cubic\fR \fIn\fR
.sp
\fB::math::figurate::biquadratic\fR \fIn\fR
.sp
\fB::math::figurate::centeredTriangular\fR \fIn\fR
.sp
\fB::math::figurate::centeredSquare\fR \fIn\fR
.sp
\fB::math::figurate::centeredPentagonal\fR \fIn\fR
.sp
\fB::math::figurate::centeredHexagonal\fR \fIn\fR
.sp
\fB::math::figurate::centeredCube\fR \fIn\fR
.sp
\fB::math::figurate::decagonal\fR \fIn\fR
.sp
\fB::math::figurate::heptagonal\fR \fIn\fR
.sp
\fB::math::figurate::hexagonal\fR \fIn\fR
.sp
\fB::math::figurate::octagonal\fR \fIn\fR
.sp
\fB::math::figurate::octahedral\fR \fIn\fR
.sp
\fB::math::figurate::pentagonal\fR \fIn\fR
.sp
\fB::math::figurate::squarePyramidral\fR \fIn\fR
.sp
\fB::math::figurate::tetrahedral\fR \fIn\fR
.sp
\fB::math::figurate::pentatope\fR \fIn\fR
.sp
.BE
.SH DESCRIPTION
.PP
Sums of numbers that follow a particular pattern are called figurate numbers\&.
A simple example is the sum of integers 1, 2, \&.\&.\&. up to n\&. You can arrange
1, 1+2=3, 1+2+3=6, \&.\&.\&. objects in a triangle, hence the name triangular
numbers:
.CS


       *
       *  *
       *  *  *
       *  *  *  *
       \&.\&.\&.

.CE
The \fBmath::figurate\fR package consists of a collection of procedures to
evaluate a wide variety of figurate numbers\&. While all formulae are
straightforward, the details are sometimes puzzling\&.
\fINote:\fR The procedures consider arguments lower than zero as to mean "no objects to be counted"
and therefore return 0\&.
.SH PROCEDURES
The procedures can be arranged in a few categories: sums of integers raised to a
particular power, sums of odd integers and general figurate numbers, for
instance the pentagonal numbers\&.
.TP
\fB::math::figurate::sum_sequence\fR \fIn\fR
Return the sum of integers 1, 2, \&.\&.\&., n\&.
.RS
.TP
int \fIn\fR
Highest integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_squares\fR \fIn\fR
Return the sum of squares 1**2, 2**2, \&.\&.\&., n**2\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_cubes\fR \fIn\fR
Return the sum of cubes 1**3, 2**3, \&.\&.\&., n**3\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_4th_power\fR \fIn\fR
Return the sum of 4th powers 1**4, 2**4, \&.\&.\&., n**4\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_5th_power\fR \fIn\fR
Return the sum of 5th powers 1**5, 2**5, \&.\&.\&., n**5\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_6th_power\fR \fIn\fR
Return the sum of 6th powers 1**6, 2**6, \&.\&.\&., n**6\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_7th_power\fR \fIn\fR
Return the sum of 7th powers 1**7, 2**7, \&.\&.\&., n**7\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.TP
\fB::math::figurate::sum_8th_power\fR \fIn\fR
Return the sum of 8th powers 1**8, 2**8, \&.\&.\&., n**8\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_9th_power\fR \fIn\fR
Return the sum of 9th powers 1**9, 2**9, \&.\&.\&., n**9\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_10th_power\fR \fIn\fR
Return the sum of 10th powers 1**10, 2**10, \&.\&.\&., n**10\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_sequence_odd\fR \fIn\fR
Return the sum of odd integers 1, 3, \&.\&.\&., 2n-1
.RS
.TP
int \fIn\fR
Highest integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_squares_odd\fR \fIn\fR
Return the sum of odd squares 1**2, 3**2, \&.\&.\&., (2n-1)**2\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_cubes_odd\fR \fIn\fR
Return the sum of odd cubes 1**3, 3**3, \&.\&.\&., (2n-1)**3\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_4th_power_odd\fR \fIn\fR
Return the sum of odd 4th powers 1**4, 2**4, \&.\&.\&., (2n-1)**4\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_5th_power_odd\fR \fIn\fR
Return the sum of odd 5th powers 1**5, 2**5, \&.\&.\&., (2n-1)**5\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_6th_power_odd\fR \fIn\fR
Return the sum of odd 6th powers 1**6, 2**6, \&.\&.\&., (2n-1)**6\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_7th_power_odd\fR \fIn\fR
Return the sum of odd 7th powers 1**7, 2**7, \&.\&.\&., (2n-1)**7\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.TP
\fB::math::figurate::sum_8th_power_odd\fR \fIn\fR
Return the sum of odd 8th powers 1**8, 2**8, \&.\&.\&., (2n-1)**8\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_9th_power_odd\fR \fIn\fR
Return the sum of odd 9th powers 1**9, 2**9, \&.\&.\&., (2n-1)**9\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::sum_10th_power_odd\fR \fIn\fR
Return the sum of odd 10th powers 1**10, 2**10, \&.\&.\&., (2n-1)**10\&.
.RS
.TP
int \fIn\fR
Highest base integer in the sum
.RE
.sp
.TP
\fB::math::figurate::oblong\fR \fIn\fR
Return the nth oblong number (twice the nth triangular number)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::pronic\fR \fIn\fR
Return the nth pronic number (synonym for oblong)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::triangular\fR \fIn\fR
Return the nth triangular number
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::square\fR \fIn\fR
Return the nth square number
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::cubic\fR \fIn\fR
Return the nth cubic number
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::biquadratic\fR \fIn\fR
Return the nth biquaratic number (i\&.e\&. n**4)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::centeredTriangular\fR \fIn\fR
Return the nth centered triangular number (items arranged in concentric
squares)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::centeredSquare\fR \fIn\fR
Return the nth centered square number (items arranged in concentric squares)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::centeredPentagonal\fR \fIn\fR
Return the nth centered pentagonal number (items arranged in concentric
pentagons)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::centeredHexagonal\fR \fIn\fR
Return the nth centered hexagonal number (items arranged in concentric
hexagons)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::centeredCube\fR \fIn\fR
Return the nth centered cube number (items arranged in concentric
cubes)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::decagonal\fR \fIn\fR
Return the nth decagonal number (items arranged in decagons with one common
vertex)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::heptagonal\fR \fIn\fR
Return the nth heptagonal number (items arranged in heptagons with one
common vertex)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::hexagonal\fR \fIn\fR
Return the nth hexagonal number (items arranged in hexagons with one
common vertex)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::octagonal\fR \fIn\fR
Return the nth octagonal number (items arranged in octagons with one
common vertex)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::octahedral\fR \fIn\fR
Return the nth octahedral number (items arranged in octahedrons with a
common centre)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::pentagonal\fR \fIn\fR
Return the nth pentagonal number (items arranged in pentagons with one
common vertex)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::squarePyramidral\fR \fIn\fR
Return the nth square pyramidral number (items arranged in a square pyramid)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::tetrahedral\fR \fIn\fR
Return the nth tetrahedral number (items arranged in a triangular pyramid)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.TP
\fB::math::figurate::pentatope\fR \fIn\fR
Return the nth pentatope number (items arranged in the four-dimensional analogue
of a triangular pyramid)
.RS
.TP
int \fIn\fR
Required index
.RE
.sp
.PP
.SH "BUGS, IDEAS, FEEDBACK"
This document, and the package it describes, will undoubtedly contain
bugs and other problems\&.
Please report such in the category \fImath :: figurate\fR of the
\fITcllib Trackers\fR [http://core\&.tcl\&.tk/tcllib/reportlist]\&.
Please also report any ideas for enhancements you may have for either
package and/or documentation\&.
.PP
When proposing code changes, please provide \fIunified diffs\fR,
i\&.e the output of \fBdiff -u\fR\&.
.PP
Note further that \fIattachments\fR are strongly preferred over
inlined patches\&. Attachments can be made by going to the \fBEdit\fR
form of the ticket immediately after its creation, and then using the
left-most button in the secondary navigation bar\&.
.SH KEYWORDS
figurate numbers, mathematics
.SH CATEGORY
Mathematics