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Figurate number — a very short introduction

KEYWORDS: Polygonal number, orthotopic number, Eulerian polynomials

Polygonal Numbers
-Pk(n) for k < 0; Pk(n) for k ≥ 0.
k\n 0 1 2 3 4 5 6  
-5 0 −1 5 18 38 65 99
-4 0 −1 4 15 32 55 84  
-3 0 −1 3 12 26 45 69  
-2 0 −1 2 9 20 35 54 A014107
-1 0 −1 1 6 14 25 39 A095794
0 0 1 0 −3 −8 −15 −24 A005563 A013648
A067998 A082562
A131386 A132411
1 0 1 1 0 −2 −5 −9 A000096 A080956
A132337 A161680
A175631
2 0 1 2 3 4 5 6 A001477
3 0 1 3 6 10 15 21 A000217, A089594
4 0 1 4 9 16 25 36 A000290 A162395
5 0 1 5 12 22 35 51 A000326

Polygonal number

The history of the figurate numbers goes back to the use of counting boards where small stones were used to represent numbers. Obviously it was helpful to arrange the stones in a figurative way, to make little pictograms for numbers.

Citing from Karl Menninger, "Number words and number symbols":

The Greeks called the counting board or table abakion, ..., the counters were called psephoi (pebbles).

In addition to placing them in the columns of the counting board, the Greeks also arranged counters independently to form 'geometrical numbers' — triangles, right angles, squares, etc.

These 'pebble-numbers' (counter-numbers), which turn up later in the history of mathematics as 'figurative numbers', thus represent the oldest association of numbers with geometrical shapes.

The triangular numbers are defined for n = 1,2,3,.. as

d_n = \sum_{i=1}^{n} i,

the square numbers as

q_n = \sum_{i=1}^{n} (2i - 1),

the pentagonal numbers as

p_n = \sum_{i=1}^{n} (3i - 2).

In general, the polygonal numbers are defined as

P_n^{k} = \sum_{i=1}^{n} ((k-2)i - (k-3)).

The sequence of partial sums of P_n^{k} (k fixed) are the pyramidal numbers. These numbers can be represented as spatial figurations.

Polygonal numbers are arithmetic sequences of order 2, pyramidal numbers are arithmetic sequences of order 3, ... In general, a figurate number of dimension r is a member of an arithmetic sequence of order r.

The roman geometers Epaphroditus and Vitrius Rufus (circa 150 AD) found the pyramidal number formula

 \sum_{i=1}^{n} P_i^{k} = (2P_n^{k} + n)(n+1)/6.

Fermat claimed to have a proof for the following proposition, which is a generalization of a theorem of Lagrange and of some interest in number theory, the polygonal number theorem:

Assume k > 2. Every integer n > 0 can be written as a sum of at most k polygonal numbers.

The theorem was proved by Legendre and Cauchy.


Author of the plots: Stefan Friedrich Birkner, License: Creative Commons Attribution-ShareAlike 3.0 Unported.

A recommended introduction to polygonal numbers is: Dan MacKinnon, Triangulating Polygonal Numbers, Mathematics Teacher, Vol. 103, No. 7, March 2010.

Orthotopic number

The generating function of the regular orthotopic numbers is given essentially by the Eulerian polynomials. This was established by Euler in 1749.

To disambiguate: Orthotopic can mean in the normal or usual position. For instance it is used in clinical medicine in this sens. Another meaning is derived from parallelotope which is called an orthotope if the spanning vectors are mutually perpendicular. In this article it is used in the latter sens.

With Maple the Eulerian Polynomials (as defined by D. E. Knuth and now also by NIST Digital Library of Mathematical Functions) can be computed as:

A := proc(n,x) if n = 0 then 1 
else x*(1-x)*diff(A(n-1,x),x)+A(n-1,x)*(1+(n-1)*x) fi end:

seq(print(sort(expand(A(n,x)))),n=0..5);

              1
              1
            x + 1
         x^2  + 4 x + 1
    x^3  + 11 x^2  + 11 x + 1
 x^4  + 26 x^3  + 66 x^2  + 26 x + 1

The generating function of the regular orthotopic numbers is given by G(n,x) = xA(n, x)/(1−x)n+1. Looking at the unshifted version F(n,x) = A(n,  x)/(1−x)n+1 we find:

F := proc(n,x) A(n,x)/(1-x)^(n+1) end:

To compute the table below:
seq(print([n^k],sort(factor(F(k,x))),
seq(coeff(series(F(k,x),x,12),x,j),j=0..5)),k=0..5);

[n^0]: (1-x)^(-1),                        1, 1,  1,   1,   1,   1
[n^1]: (1-x)^(-2),                        1, 2,  3,   4,   5,   6
[n^2]: (1-x)^(-3)(x+1),                   1, 4,  9,  16,  25,  36
[n^3]: (1-x)^(-4)(x^2+4x+1),              1, 8, 27,  64, 125, 216
[n^4]: (1-x)^(-5)(x+1)(x^2+10x+1),        1,16, 81, 256, 625,1296
[n^5]: (1-x)^(-6)(x^4+26x^3+66x^2+26x+1), 1,32,243,1024,3125,7776

Reading the rectangular array by diagonals in western style (left to right) gives the

Triangle of orthotopic numbers  A009998
1  
11  
121  
1431  
18941  
116271651  
13281642561

Peter Bala observed that this is the Hilbert transform of the Eulerian numbers A123125. (See A145905 for the definition of the Hilbert transform of a triangular array.) Note that this is just a rephrasing of Euler's 1749 theorem. Unfortunately several different conventions with regard to the enumeration of the Eulerian numbers are in use. Bala remarks: 'The triangle of Eulerian numbers A008292 [can be viewed] as the coefficients of h-polynomials of n-dimensional permutohedra of type A).' (See also the literature cited by Bala in A145905.)



The generating function of the centered regular orthotopic numbers is given by

H := proc(n,x) (1+x)*A(n,x)/(1-x)^(n+1) end:

To compute the table below:
seq(print([n^k],sort(factor(H(k,x))),
seq(coeff(series(H(k,x),x,12),x,j),j=0..5)),k=0..5);

[n^0]: (1-x)^(-1)(1+x),                        1, 2,  2,   2,   2,    2
[n^1]: (1-x)^(-2)(1+x),                        1, 3,  5,   7,   9,   11
[n^2]: (1-x)^(-3)(1+x)(x+1),                   1, 5, 13,  25,  41,   61
[n^3]: (1-x)^(-4)(1+x)(x^2+4x+1),              1, 9, 35,  91, 189,  341
[n^4]: (1-x)^(-5)(1+x)(x+1)(x^2+10x+1),        1,17, 97, 337, 881, 1921
[n^5]: (1-x)^(-6)(1+x)(x^4+26x^3+66x^2+26x+1), 1,33,275,1267,4149,10901

Reading the rectangular array by diagonals in western style (left to right) gives the

Triangle of centered orthotopic numbers   A179927
1  
12  
132  
1552  
191372  
117352592  
133979141112

Note that this triangle is related to the triangle A008518. In a more formal setup the exact wording depends on the definition of the Eulerian polynomials. Row sums are in A179928.

A link-farm to the database corresponding to the rows of the two numerical rectangles above. (Note that the offset of the sequences possibly needs to be adjusted to meet your conventions.)

GF 0 1 2 3 4 5
A(n,x)x/(1-x)^(n+1) A057427 A001477 A000290 A000578 A000583 A000584
A(n,x)(x+1)/(1-x)^(n+1) A040000 A005408 A001844 A005898 A008514 A008515



See the impressive contribution of Daniel Forgues to the Classifications of figurate numbers. A recommended reading is Regular polytopes by H. S. M. Coxeter, Dover Publications.

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