Polytopic Numbers and Generating Functions

Tetrahedral numbers: related to the tetrahedron $1; 4; 10; 20; 35; 56; 84; \ldots$ Octahedral numbers: related to the octahedron $1; 6; 19; 44; 85; 146; \ldots$ Cubic numbers: related to the cube $1; 8; 27; 64; 125; 216; \ldots$ Icosahedral numbers: related to the icosahedron $1; 12; 48; 124; 255; 456; \ldots$ Dodecahedral numbers: related to the dodecahedron $1; 20; 84; 220; 455; 816; \ldots$ through which it is possible to form a pyramid whose base can be any regular polygon. The general formula for tetrahedral numbers is $\frac{1}{6}n(n+1)(n+2)$ The general formula for octahedral numbers is $\frac{1}{3}n(2n^2+1)$ The general formula for cubic numbers is $n^3$ The general formula for icosahedral numbers is $\frac{1}{2}n(5n^2-5n+2)$ The general formula for dodecahedral numbers is $\frac{1}{2}n(3n-1)(3n-2)$ Table of Polygonal Numbers. Anyone who has studied elements of plane geometry knows that the triangular figure can be considered as a kind of fundamental geometric atom. Indeed, no plane figure exists with fewer vertices, and the coordination of multiple triangles gives rise to all regular polygons. More generally, all plane geometric figures are triangularizable. The same geometric situation finds its arithmetic analogue in the table we will construct and which we have called the table of polygonal numbers. The set of polygonal numbers can be easily generated starting from the series of triangular numbers conjugated to the sequence of natural numbers. Let us consider, indeed, the sequence of triangular numbers which in this context represents our key row: $T(n): 1; 3; 6; 10; 15; 21; 28; \ldots$ By shifting this sequence by one term and adding it to the natural numbers, we obtain again the series of triangular numbers: triangular: $1; 3; 6; 10; 15; 21; \ldots$ + natural: $1; 2; 3; 4; 5; 6; 7; \ldots$ = triangular: $1; 3; 6; 10; 15; 21; 28; \ldots$ To obtain the sequence of square numbers $\{1; 4; 9; 16; \ldots\}$, it is sufficient to add again the key row of triangular numbers: triangular: $1; 3; 6; 10; 15; 21; \ldots$ + triangular: $1; 3; 6; 10; 15; 21; 28; \ldots$ = squares: $1; 4; 9; 16; 25; 36; 49; \ldots$ Adding again the sequence of triangular numbers we obtain the pentagonal numbers: triangular: $1; 3; 6; 10; 15; 21; \ldots$ + squares: $1; 4; 9; 16; 25; 36; 49; \ldots$ = pentagonal: $1; 5; 12; 22; 35; 51; 70; \ldots$ and so on we can obtain all polygonal numbers by constructing, thanks to the repeated sum of triangular numbers, what we have called the table of polygonal numbers. Simplicial key of polygonal numbers $T(n): 1, 3, 6, 10, 15, 21, 28, \ldots$ The key row is to be added to each row to construct the next one. Algebraically this property can be described considering that every number $m$, $n$-th polygonal number, relative to a polygon of $q$ sides can be written as the sum: $m = pol_q(n) = T(n) + (q-3)T(n-1)$ Generating Functions and Figurate Numbers. An interesting way to treat sequences of natural numbers is to associate the sequence with a generating function. Despite the name, generating functions are generally not functions, but formal power series whose coefficients are constituted by the elements of the original sequence. For example: if we consider the Fibonacci sequence $\{1; 1; 2; 3; 5; 8; 13; 21; \ldots; F_n; \ldots\}$, we can associate to this sequence the formal series or generating function: $\Phi(z) = 1 + z + 2z^2 + 3z^3 + 5z^4 + 8z^5 + \cdots = \sum_{n=1}^{\infty} F_n z^n$ From a purely practical point of view, the advantage of this association comes out when we want to know the successive terms of the sequence without knowing the law that generates them. If, indeed, we could see that the generating function $\Phi(z)$ is nothing but the power series expansion at the origin of the function: $\Phi(z) = \sum_{n=1}^{\infty} F_n z^n = \frac{1}{1-z-z^2}$ we could then continue the series development and obtain all the necessary terms of the sequence. In the same way, through the generating function, one can find a quick formula that approximates any term of the sequence: $F_n \approx \frac{1}{\sqrt{5}} \phi^n$ where $\phi = \frac{1+\sqrt{5}}{2} = 1.6180\ldots$ is the golden number or golden ratio. In this way we can quickly calculate, for example, the 22nd number of the sequence without knowing the preceding ones, i.e. $F_{22} = \frac{177111}{\sqrt{5}} \phi^{22}$. Beyond computational practicality, studying the algebraic properties and symmetries of a generating function of a sequence is generally much simpler than studying the sequence itself, and this makes it a very useful tool of contemporary mathematics. To find the generating functions of Figurate Numbers we can proceed easily through the use of two notable generating functions. The first is relative to the sequence formed entirely of units, i.e. $\{1; 1; 1; 1; \ldots\}$ which has as associated formal series: $\sum_{n=1}^{\infty} z^n = 1 + z + z^2 + z^3 + \cdots = \frac{1}{1-z}$ and therefore the generating function of the sequence will be: $\{1; 1; 1; 1; \ldots\} \rightarrow \frac{1}{1-z}$ We note one last time that from an analytical point of view the series converges only with $|z| < 1$, but that the series in question must be taken as formal series.