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(Centered squares) hyperpyramidal numbers

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The (centered squares) hyperpyramidal numbers are starting with a centered square for dimension d = 2, then do a pyramidal stacking for each extra dimension.

All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Formulae

${\displaystyle \,_{'c'}Y_{d+2}^{(d)}(n)=?\,}$ [1]

Schläfli-Poincaré (convex) polytope formula

Schläfli-Poincaré generalization of the Descartes-Euler (convex) polyhedral formula.[2]

For d-dimensional convex polytopes:

${\displaystyle {\sum _{i=0}^{d-1}(-1)^{i}N_{i}}=1-(-1)^{d},\,}$

where N0 is the number of 0-dimensional elements (vertices V), N1 is the number of 1-dimensional elements (edges V), N2 is the number of 2-dimensional elements (faces F) and N3 is the number of 3-dimensional elements (cells C), and so on... of the convex polytope.

Recurrence equation

${\displaystyle \,_{'c'}Y_{d+2}^{(d)}(n)=?,\,}$
${\displaystyle \,_{'c'}Y_{d+2}^{(d)}(0)=1,\ \,_{'c'}Y_{d+2}^{(d)}(1)=?,\ \,_{'c'}Y_{d+2}^{(d)}(2)=?.\,}$

Generating function

${\displaystyle G_{\{\,_{'c'}Y_{d+2}^{(d)}\}}(x)=?\,}$

Order of basis

${\displaystyle g_{\{\,_{'c'}Y_{d+2}^{(d)}\}}=?\,}$

Differences

${\displaystyle \,_{'c'}Y_{d+2}^{(d)}(n)-\,_{'c'}Y_{d+2}^{(d)}(n-1)=?\,}$

Partial sums

${\displaystyle \sum _{n=1}^{m}\,_{'c'}Y_{d+2}^{(d)}(n)=\,}$

Partial sums of reciprocals

${\displaystyle \sum _{n=1}^{m}{\frac {1}{\,_{'c'}Y_{d+2}^{(d)}(n)}}=\,}$

Sum of reciprocals

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{\,_{'c'}Y_{d+2}^{(d)}(n)}}=\,}$

Table of formulae and values

N0, N1, N2,N3, ... are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the (n-1)-dimensional "vertices" are the actual facets. The (centered squares) hyperpyramidal numbers are listed by increasing number N0 of vertices.

(Centered squares) hyperpyramidal numbers formulae and values
d Name

(N0, N1, N2, ...)

Schläfli symbol[3]

Formulae

${\displaystyle \,_{'c'}Y_{d+2}^{(d)}(n)=\,}$

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 OEIS

number

3 Centered square pyramidal

(, , )

{, }

${\displaystyle {\frac {n(2n^{2}+1)}{3}}\,}$

${\displaystyle P_{2\cdot 3}^{(3)}(n)\,}$

0 1 6 19 44 85 146 231 344 489 670 891 1156 A005900(n)
4

(, , , )

{, , }

${\displaystyle \,}$ 1
5

(, , , , )

{, , , }

${\displaystyle \,}$ 1
6

(, , , , , )

{, , , , }

${\displaystyle \,}$ 1
7

(, , , , , , )

{, , , , , }

${\displaystyle \,}$ 1
8

(, , , , , , , )

{, , , , , , }

${\displaystyle \,}$ 1
9

(, , , , , , , , )

{, , , , , , , }

${\displaystyle \,}$ 1
10

(, , , , , , , , , )

{, , , , , , , , }

${\displaystyle \,}$ 1
11

(, , , , , , , , , , )

{, , , , , , , , , }

${\displaystyle \,}$ 1
12

(, , , , , , , , , , , )

{, , , , , , , , , , }

${\displaystyle \,}$ 1

Table of related formulae and values

N0, N1, N2,N3, ... are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the (n-1)-dimensional elements are the actual facets. The (centered squares) hyperpyramidal numbers are listed by increasing number N0 of vertices.

(Centered squares) hyperpyramidal numbers related formulae and values
d Name

(N0, N1, N2, ...)

Schläfli symbol[3]

Generating

function

${\displaystyle G_{\{\,_{'c'}Y_{d+2}^{(d)}\}}(x)\,}$

Order

of basis

Differences

${\displaystyle \,_{'c'}Y_{d+2}^{(d)}(n)-\,}$

${\displaystyle \,_{'c'}Y_{d+2}^{(d)}(n-1)=\,}$

${\displaystyle \,_{'c'}Y_{d+1}^{(d-1)}(n)\,}$

Partial sums

${\displaystyle \sum _{n=1}^{m}{\,_{'c'}Y_{d+2}^{(d)}(n)}=}$

${\displaystyle \,_{'c'}Y_{d+3}^{(d+1)}(m)\,}$

Partial sums of reciprocals

${\displaystyle \sum _{n=1}^{m}{1 \over {\,_{'c'}Y_{d+2}^{(d)}(n)}}=}$

Sum of reciprocals[7]

${\displaystyle \sum _{n=1}^{\infty }{1 \over {\,_{'c'}Y_{d+2}^{(d)}(n)}}=}$

3 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
4 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
5 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
6 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
7 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
8 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
9 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
10 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
11 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$
12 ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$ ${\displaystyle \,}$

Table of sequences

(Centered squares) hyperpyramidal numbers sequences
d ${\displaystyle \,_{'c'}Y_{d+2}^{(d)}(n),\ n\geq 0\,}$ sequences
3 {0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, ...}
4 {0, 1, ...}
5 {0, 1, ...}
6 {0, 1, ...}
7 {0, 1, ...}
8 {0, 1, ...}
9 {0, 1, ...}
10 {0, 1, ...}
11 {0, 1, ...}
12 {0, 1, ...}

1. Where ${\displaystyle \scriptstyle \,_{'c'}Y_{[(k+2)+(d-2)]}^{(d)}(n)=\,_{'c'}Y_{k+d}^{(d)}(n)\,}$, k ≥ 1, n ≥ 0, is the d-dimensional, d ≥ 0, (centered (k+2)-gonal base) (hyper)pyramidal number where, for d ≥ 2, ${\displaystyle \scriptstyle N_{0}=[(k+2)+(d-2)]\,}$ is the number of vertices (including the ${\displaystyle \scriptstyle d-2\,}$ apex vertices) of the (centered polygonal base) (hyper)pyramid (the quoted ${\displaystyle \scriptstyle 'c'\,}$ emphasizes that only the polygons are centered, not the whole figure.)