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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, ...}

Notes

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.)
2. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
3. Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
4. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
5. HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
6. Pollock, Frederick, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders, Abstracts of the Papers Communicated to the Royal Society of London, 5 (1850) pp. 922-924.
7. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.