math::figurate - Evaluate figurate numbers

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Description

       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:

                     *
                     *  *
                     *  *  *
                     *  *  *  *
                     ...

       The  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.   Note:  The
       procedures  consider arguments lower than zero as to mean "no objects to be counted" and therefore return
       0.

Keywords

       figurate numbers, mathematics

Name

       math::figurate - Evaluate figurate numbers

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.

       ::math::figurate::sum_sequencen
              Return the sum of integers 1, 2, ..., n.

              int n  Highest integer in the sum

       ::math::figurate::sum_squaresn
              Return the sum of squares 1**2, 2**2, ..., n**2.

              int n  Highest base integer in the sum

       ::math::figurate::sum_cubesn
              Return the sum of cubes 1**3, 2**3, ..., n**3.

              int n  Highest base integer in the sum

       ::math::figurate::sum_4th_powern
              Return the sum of 4th powers 1**4, 2**4, ..., n**4.

              int n  Highest base integer in the sum

       ::math::figurate::sum_5th_powern
              Return the sum of 5th powers 1**5, 2**5, ..., n**5.

              int n  Highest base integer in the sum

       ::math::figurate::sum_6th_powern
              Return the sum of 6th powers 1**6, 2**6, ..., n**6.

              int n  Highest base integer in the sum

       ::math::figurate::sum_7th_powern
              Return the sum of 7th powers 1**7, 2**7, ..., n**7.

              int n  Highest base integer in the sum

       ::math::figurate::sum_8th_powern
              Return the sum of 8th powers 1**8, 2**8, ..., n**8.

              int n  Highest base integer in the sum

       ::math::figurate::sum_9th_powern
              Return the sum of 9th powers 1**9, 2**9, ..., n**9.

              int n  Highest base integer in the sum

       ::math::figurate::sum_10th_powern
              Return the sum of 10th powers 1**10, 2**10, ..., n**10.

              int n  Highest base integer in the sum

       ::math::figurate::sum_sequence_oddn
              Return the sum of odd integers 1, 3, ..., 2n-1

              int n  Highest integer in the sum

       ::math::figurate::sum_squares_oddn
              Return the sum of odd squares 1**2, 3**2, ..., (2n-1)**2.

              int n  Highest base integer in the sum

       ::math::figurate::sum_cubes_oddn
              Return the sum of odd cubes 1**3, 3**3, ..., (2n-1)**3.

              int n  Highest base integer in the sum

       ::math::figurate::sum_4th_power_oddn
              Return the sum of odd 4th powers 1**4, 2**4, ..., (2n-1)**4.

              int n  Highest base integer in the sum

       ::math::figurate::sum_5th_power_oddn
              Return the sum of odd 5th powers 1**5, 2**5, ..., (2n-1)**5.

              int n  Highest base integer in the sum

       ::math::figurate::sum_6th_power_oddn
              Return the sum of odd 6th powers 1**6, 2**6, ..., (2n-1)**6.

              int n  Highest base integer in the sum

       ::math::figurate::sum_7th_power_oddn
              Return the sum of odd 7th powers 1**7, 2**7, ..., (2n-1)**7.

              int n  Highest base integer in the sum

       ::math::figurate::sum_8th_power_oddn
              Return the sum of odd 8th powers 1**8, 2**8, ..., (2n-1)**8.

              int n  Highest base integer in the sum

       ::math::figurate::sum_9th_power_oddn
              Return the sum of odd 9th powers 1**9, 2**9, ..., (2n-1)**9.

              int n  Highest base integer in the sum

       ::math::figurate::sum_10th_power_oddn
              Return the sum of odd 10th powers 1**10, 2**10, ..., (2n-1)**10.

              int n  Highest base integer in the sum

       ::math::figurate::oblongn
              Return the nth oblong number (twice the nth triangular number)

              int n  Required index

       ::math::figurate::pronicn
              Return the nth pronic number (synonym for oblong)

              int n  Required index

       ::math::figurate::triangularn
              Return the nth triangular number

              int n  Required index

       ::math::figurate::squaren
              Return the nth square number

              int n  Required index

       ::math::figurate::cubicn
              Return the nth cubic number

              int n  Required index

       ::math::figurate::biquadraticn
              Return the nth biquaratic number (i.e. n**4)

              int n  Required index

       ::math::figurate::centeredTriangularn
              Return the nth centered triangular number (items arranged in concentric squares)

              int n  Required index

       ::math::figurate::centeredSquaren
              Return the nth centered square number (items arranged in concentric squares)

              int n  Required index

       ::math::figurate::centeredPentagonaln
              Return the nth centered pentagonal number (items arranged in concentric pentagons)

              int n  Required index

       ::math::figurate::centeredHexagonaln
              Return the nth centered hexagonal number (items arranged in concentric hexagons)

              int n  Required index

       ::math::figurate::centeredCuben
              Return the nth centered cube number (items arranged in concentric cubes)

              int n  Required index

       ::math::figurate::decagonaln
              Return the nth decagonal number (items arranged in decagons with one common vertex)

              int n  Required index

       ::math::figurate::heptagonaln
              Return the nth heptagonal number (items arranged in heptagons with one common vertex)

              int n  Required index

       ::math::figurate::hexagonaln
              Return the nth hexagonal number (items arranged in hexagons with one common vertex)

              int n  Required index

       ::math::figurate::octagonaln
              Return the nth octagonal number (items arranged in octagons with one common vertex)

              int n  Required index

       ::math::figurate::octahedraln
              Return the nth octahedral number (items arranged in octahedrons with a common centre)

              int n  Required index

       ::math::figurate::pentagonaln
              Return the nth pentagonal number (items arranged in pentagons with one common vertex)

              int n  Required index

       ::math::figurate::squarePyramidraln
              Return the nth square pyramidral number (items arranged in a square pyramid)

              int n  Required index

       ::math::figurate::tetrahedraln
              Return the nth tetrahedral number (items arranged in a triangular pyramid)

              int n  Required index

       ::math::figurate::pentatopen
              Return  the  nth pentatope number (items arranged in the four-dimensional analogue of a triangular
              pyramid)

              int n  Required index

Synopsis

       package require Tcl8.69

       package require math::figurate1.0::math::figurate::sum_sequencen::math::figurate::sum_squaresn::math::figurate::sum_cubesn::math::figurate::sum_4th_powern::math::figurate::sum_5th_powern::math::figurate::sum_6th_powern::math::figurate::sum_7th_powern::math::figurate::sum_8th_powern::math::figurate::sum_9th_powern::math::figurate::sum_10th_powern::math::figurate::sum_sequence_oddn::math::figurate::sum_squares_oddn::math::figurate::sum_cubes_oddn::math::figurate::sum_4th_power_oddn::math::figurate::sum_5th_power_oddn::math::figurate::sum_6th_power_oddn::math::figurate::sum_7th_power_oddn::math::figurate::sum_8th_power_oddn::math::figurate::sum_9th_power_oddn::math::figurate::sum_10th_power_oddn::math::figurate::oblongn::math::figurate::pronicn::math::figurate::triangularn::math::figurate::squaren::math::figurate::cubicn::math::figurate::biquadraticn::math::figurate::centeredTriangularn::math::figurate::centeredSquaren::math::figurate::centeredPentagonaln::math::figurate::centeredHexagonaln::math::figurate::centeredCuben::math::figurate::decagonaln::math::figurate::heptagonaln::math::figurate::hexagonaln::math::figurate::octagonaln::math::figurate::octahedraln::math::figurate::pentagonaln::math::figurate::squarePyramidraln::math::figurate::tetrahedraln::math::figurate::pentatopen

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