[manpage_begin math::figurate n 1.0] [keywords {figurate numbers}] [keywords mathematics] [moddesc {Tcl Math Library}] [titledesc {Evaluate figurate numbers}] [category Mathematics] [require Tcl 8.6] [require math::figurate 1.0] [description] [para] 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: [example { * * * * * * * * * * ... }] The [package math::figurate] 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. [emph Note:] The procedures consider arguments lower than zero as to mean "no objects to be counted" and therefore return 0. [section "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. [list_begin definitions] [call [cmd ::math::figurate::sum_sequence] [arg n]] Return the sum of integers 1, 2, ..., n. [list_begin arguments] [arg_def int n] Highest integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_squares] [arg n]] Return the sum of squares 1**2, 2**2, ..., n**2. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_cubes] [arg n]] Return the sum of cubes 1**3, 2**3, ..., n**3. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_4th_power] [arg n]] Return the sum of 4th powers 1**4, 2**4, ..., n**4. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_5th_power] [arg n]] Return the sum of 5th powers 1**5, 2**5, ..., n**5. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_6th_power] [arg n]] Return the sum of 6th powers 1**6, 2**6, ..., n**6. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_7th_power] [arg n]] Return the sum of 7th powers 1**7, 2**7, ..., n**7. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [call [cmd ::math::figurate::sum_8th_power] [arg n]] Return the sum of 8th powers 1**8, 2**8, ..., n**8. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_9th_power] [arg n]] Return the sum of 9th powers 1**9, 2**9, ..., n**9. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_10th_power] [arg n]] Return the sum of 10th powers 1**10, 2**10, ..., n**10. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_sequence_odd] [arg n]] Return the sum of odd integers 1, 3, ..., 2n-1 [list_begin arguments] [arg_def int n] Highest integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_squares_odd] [arg n]] Return the sum of odd squares 1**2, 3**2, ..., (2n-1)**2. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_cubes_odd] [arg n]] Return the sum of odd cubes 1**3, 3**3, ..., (2n-1)**3. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_4th_power_odd] [arg n]] Return the sum of odd 4th powers 1**4, 2**4, ..., (2n-1)**4. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_5th_power_odd] [arg n]] Return the sum of odd 5th powers 1**5, 2**5, ..., (2n-1)**5. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_6th_power_odd] [arg n]] Return the sum of odd 6th powers 1**6, 2**6, ..., (2n-1)**6. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_7th_power_odd] [arg n]] Return the sum of odd 7th powers 1**7, 2**7, ..., (2n-1)**7. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [call [cmd ::math::figurate::sum_8th_power_odd] [arg n]] Return the sum of odd 8th powers 1**8, 2**8, ..., (2n-1)**8. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_9th_power_odd] [arg n]] Return the sum of odd 9th powers 1**9, 2**9, ..., (2n-1)**9. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::sum_10th_power_odd] [arg n]] Return the sum of odd 10th powers 1**10, 2**10, ..., (2n-1)**10. [list_begin arguments] [arg_def int n] Highest base integer in the sum [list_end] [para] [call [cmd ::math::figurate::oblong] [arg n]] Return the nth oblong number (twice the nth triangular number) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::pronic] [arg n]] Return the nth pronic number (synonym for oblong) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::triangular] [arg n]] Return the nth triangular number [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::square] [arg n]] Return the nth square number [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::cubic] [arg n]] Return the nth cubic number [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::biquadratic] [arg n]] Return the nth biquaratic number (i.e. n**4) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::centeredTriangular] [arg n]] Return the nth centered triangular number (items arranged in concentric squares) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::centeredSquare] [arg n]] Return the nth centered square number (items arranged in concentric squares) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::centeredPentagonal] [arg n]] Return the nth centered pentagonal number (items arranged in concentric pentagons) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::centeredHexagonal] [arg n]] Return the nth centered hexagonal number (items arranged in concentric hexagons) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::centeredCube] [arg n]] Return the nth centered cube number (items arranged in concentric cubes) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::decagonal] [arg n]] Return the nth decagonal number (items arranged in decagons with one common vertex) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::heptagonal] [arg n]] Return the nth heptagonal number (items arranged in heptagons with one common vertex) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::hexagonal] [arg n]] Return the nth hexagonal number (items arranged in hexagons with one common vertex) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::octagonal] [arg n]] Return the nth octagonal number (items arranged in octagons with one common vertex) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::octahedral] [arg n]] Return the nth octahedral number (items arranged in octahedrons with a common centre) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::pentagonal] [arg n]] Return the nth pentagonal number (items arranged in pentagons with one common vertex) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::squarePyramidral] [arg n]] Return the nth square pyramidral number (items arranged in a square pyramid) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::tetrahedral] [arg n]] Return the nth tetrahedral number (items arranged in a triangular pyramid) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [call [cmd ::math::figurate::pentatope] [arg n]] Return the nth pentatope number (items arranged in the four-dimensional analogue of a triangular pyramid) [list_begin arguments] [arg_def int n] Required index [list_end] [para] [list_end] [vset CATEGORY {math :: figurate}] [include ../common-text/feedback.inc] [manpage_end]