A276985
Triangle read by rows: T(n,k) = number of k-dimensional elements in an n-dimensional cross-polytope, n>=1, 0<=k
2, 4, 4, 6, 12, 8, 8, 24, 32, 16, 10, 40, 80, 80, 32, 12, 60, 160, 240, 192, 64, 14, 84, 280, 560, 672, 448, 128, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120
Offset: 1
Examples
T(4, 1..4) = 8, 24, 32, 16, because the 16-cell has 8 0-faces (vertices), 24 1-faces (edges), 32 2-faces (faces) and 16 3-faces (cells). Triangle starts 2 4, 4 6, 12, 8 8, 24, 32, 16 10, 40, 80, 80, 32 12, 60, 160, 240, 192, 64 14, 84, 280, 560, 672, 448, 128 16, 112, 448, 1120, 1792, 1792, 1024, 256 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024
References
- H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, 1973, ISBN 9780486141589.
Links
- Wikipedia, Cross-polytope.
- D. A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 666 (2017), 21-35.
Crossrefs
Programs
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Mathematica
Table[2^(k + 1) Binomial[n, k + 1], {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Sep 25 2016 *) -
PARI
T(n, k) = 2^(k+1)*binomial(n, k+1) trianglerows(n) = for(x=1, n, for(y=0, x-1, print1(T(x, y), ", ")); print("")) trianglerows(10) \\ print initial 10 rows of triangle
Formula
T(n,k) = 2^(k+1) * binomial(n, k+1) (cf. Coxeter, 1973, formula 7.22).
G.f.: 2*x/((1 - x)*(1 - x - 2*x*y)). - Stefano Spezia, Jul 17 2025
Comments