A167986 -id:A167986 - cp's OEIS frontend

A167987 Number of (undirected) cycles in the graph of the n-orthoplex, n>=2.

1, 63, 2766, 194650, 21086055, 3257119761, 679314442828, 183842034768036, 62630787876947325, 26224409462275175635, 13236607762537219815546, 7925653200467421739217118, 5554198822066977588903819331, 4503367772662184077396436475525, 4182811121982123218357983540881240

Offset: 2

Andrew Weimholt, Nov 16 2009

nonn

Row sums of triangle in A167986.
The n-orthoplex, also known as the n-cross-polytope, is the dual of the n-cube.
A.k.a. number of (undirected) cycles in the n-cocktail party graph. - Eric W. Weisstein, Dec 29 2013

			a(3) = 63, because in dimension n=3, the orthoplex is the octahedron, which has 63 cycles in its graph.

Andrew Howroyd, Table of n, a(n) for n = 2..100
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Cross Polytope
Eric Weisstein's World of Mathematics, Graph Cycle

Cf. A167986.

b:= func< n,k,j | (-1)^j*Binomial(n,j)*Binomial(2*(n-j),k-2*j)*2^(j-1)*Factorial(k-j-1) >;
A167986:= func< n,k | (&+[b(n,k,j): j in [0..Floor(k/2)]]) >;
A167987:= func< n | (&+[A167986(n,k): k in [3..2*n]]) >;
[A167987(n): n in [2..30]]; // G. C. Greubel, Jan 17 2023

a[n_]:= Sum[Sum[(-1)^j*Binomial[n, j]*Binomial[2*(n-j), k-2*j]*2^j*(k - j-1)!, {j, 0, k/2}], {k, 3, 2 n}]/2; Array[a, 15, 2] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

a(n)=sum(k=3,2*n, sum(j=0,k\2, (-1)^j*binomial(n,j)*binomial(2*(n-j),k-2*j)*2^j*(k-j-1)!))/2; \\ Andrew Howroyd, May 09 2017

def A167986(n,k): return simplify(binomial(2*n, k)*gamma(k)*hypergeometric([(1-k)/2, -k/2], [1-k, 1/2 -n], -2)/2)
@CachedFunction
def A167987(n): return sum(A167986(n,k) for k in range(3,2*n+1))
[A167987(n) for n in range(2,31)] # G. C. Greubel, Jan 17 2023

a(n) = Sum_{k=3..2*n} Sum_{j=0..floor(k/2)} (-1)^j*binomial(n,j) * binomial(2*(n-j),k-2*j) * 2^j*(k-j-1)!/2. - Andrew Howroyd, May 09 2017

a(8)-a(11) from Eric W. Weisstein, Dec 19 2013
a(12) from Eric W. Weisstein, Dec 21 2013
a(13) from Eric W. Weisstein, Jan 08 2014
a(14) from Eric W. Weisstein, Apr 09 2014
a(15)-a(16) from Andrew Howroyd, May 09 2017

A167981 Number of 2n-cycles on the graph of the tesseract, 2 <= n <= 8.

Original entry on oeis.org

24, 128, 696, 2112, 5024, 5736, 1344

Offset: 2

Andrew Weimholt, Nov 16 2009

Row n=4 of the triangle in A085452
The graph of any n-cube (n>1) contains only even length cycles.
The tesseract is the 4 dimensional cube, and is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol for the tesseract is {4,3,3}.

			a(2) = 24 because there are 24 4-cycles on the graph of the tesseract.
The cycle polynomial is  24*x^4 + 128*x^6 + 696*x^8 + 2112*x^10 + 5024*x^12 + 5376*x^14 + 1344*x^16.

A. Weimholt, Tesseract Foldout
Eric Weisstein's World of Mathematics, Cycle Polynomial
Eric Weisstein's World of Mathematics, Tesseract Graph

Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).

A167982 Number of n-cycles on the graph of the regular 16-cell, 3 <= n <= 8.

Original entry on oeis.org

32, 102, 288, 640, 960, 744

Offset: 3

Andrew Weimholt, Nov 16 2009

Row n=3 of the triangle in A167986

The 16-cell is the dual polytope of the tesseract, and is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol for the 16-cell is {3,3,4}.

			a(3) = 32, because there are 32 3-cycles on the graph of the 16-cell.
Cycle polynomial is 32*x^3 + 102*x^4 + 288*x^5 + 640*x^6 + 960*x^7 + 744*x^8.

A. Weimholt, 16-cell net
Eric Weisstein's World of Mathematics, 16-Cell
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).

A167983 Number of n-cycles on the graph of the regular 24-cell, 3 <= n <= 24.

Original entry on oeis.org

96, 360, 1440, 7120, 37728, 196488, 974592, 4536000, 19934208, 82689264, 322437312, 1171745280, 3924079104, 11964375936, 32761139328, 79244294016, 165800420352, 291640320576, 413774810112, 443415854592, 318534709248, 114869295744

Offset: 3

Andrew Weimholt, Nov 16 2009

The 24-cell is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol of the 24-cell is {3,4,3}.

			a(3) = 96, because there are 96 3-cycles on the graph of the 24-cell.
Cycle polynomial is 96*x^3 + 360*x^4 + 1440*x^5 + 7120*x^6 + 37728*x^7 + 196488*x^8 + 974592*x^9 + 4536000*x^10 + 19934208*x^11 + 82689264*x^12 + 322437312*x^13 + 1171745280*x^14 + 3924079104*x^15 + 11964375936*x^16 + 32761139328*x^17 + 79244294016*x^18 + 165800420352*x^19 + 291640320576*x^20 + 413774810112*x^21 + 443415854592*x^22 + 318534709248*x^23 + 114869295744*x^24.

Max Alekseyev, PARI/GP scripts for various math problems
Max A. Alekseyev, Gérard P. Michon, Making Walks Count: From Silent Circles to Hamiltonian Cycles, arXiv:1602.01396 [math.CO], 2016.
A. Weimholt, 24-cell net
Eric Weisstein's World of Mathematics, 24-Cell
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).

a(16)-a(24) and "full" keyword from Max Alekseyev, Nov 18 2009

A167984 Number of n-cycles on the graph of the regular 120-cell, 3 <= n <= 600.

Original entry on oeis.org

0, 0, 720, 0, 0, 3600, 2400, 4320, 28800, 35400, 64800, 284400, 540000, 1139400, 3708000, 8557200, 19677600, 55725120, 140359200, 346456800, 935942400, 2442469200, 6282571680

Offset: 3

Andrew Weimholt, Nov 16 2009

The 120-cell is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol of the 120-cell is {5,3,3}.

			a(5) = 720, because there are 720 5-cycles on the graph of the 120-cell.
Cycle polynomial is 720*x^5 + 3600*x^8 + 2400*x^9 + 4320*x^10 + 28800*x^11 + 35400*x^12 + 64800*x^13 +  284400*x^14 + 540000*x^15 + 1139400*x^16 + 3708000*x^17 + 8557200*x^18 + 19677600*x^19 + 55725120*x^20 + 140359200*x^21 + 346456800*x^22 + 935942400*x^23 + ...

A. Weimholt, 120-cell net
Eric Weisstein's World of Mathematics, 120-Cell
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167985 (n-cycles on graph of 600-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).
Cf. A108997 (number of vertices n-steps from a given vertex on graph of 120-cell).

a(24) from Eric W. Weisstein, Feb 21 2014

a(25) from Eric W. Weisstein, Mar 11 2014

A167985 Number of n-cycles on the graph of the regular 600-cell, 3 <= n <= 120.

Original entry on oeis.org

1200, 5400, 29520, 187200, 1310400, 9813600, 77193600, 630538632, 5307656400

Offset: 3

Andrew Weimholt, Nov 16 2009

The 600-cell is one of 6 regular convex polytopes in 4 dimensions. The Schläfli symbol for the 600-cell is {3,3,5}.

			a(3) = 1200, because there are 1200 3-cycles on the graph of the 600-cell.
Cycle polynomial is 1200*x^3 + 5400*x^4 + 29520*x^5 + 187200*x^6 + 1310400*x^7 + 9813600*x^8 + 77193600*x^9 + 630538632*x^10 + ...

A. Weimholt, 600-cell net
Eric Weisstein's World of Mathematics, 600-Cell
Eric Weisstein's World of Mathematics, Cycle Polynomial

Cf. A167981 (2n-cycles on graph of the tesseract).
Cf. A167982 (n-cycles on graph of 16-cell).
Cf. A167983 (n-cycles on graph of 24-cell).
Cf. A167984 (n-cycles on graph of 120-cell).
Cf. A085452 (2k-cycles on graph of n-cube).
Cf. A144151 (ignoring first three columns (0<=k<=2), k-cycles on (n-1)-simplex).
Cf. A167986 (k-cycles on graph of n-orthoplex).
Cf. A118785 (number of vertices n-steps from a given vertex on graph of the 600-cell).

a(11) from Eric W. Weisstein, Feb 09 2014

cp's OEIS Frontend

A167987 Number of (undirected) cycles in the graph of the n-orthoplex, n>=2.

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A167981 Number of 2n-cycles on the graph of the tesseract, 2 <= n <= 8.

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A167982 Number of n-cycles on the graph of the regular 16-cell, 3 <= n <= 8.

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A167983 Number of n-cycles on the graph of the regular 24-cell, 3 <= n <= 24.

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A167984 Number of n-cycles on the graph of the regular 120-cell, 3 <= n <= 600.

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A167985 Number of n-cycles on the graph of the regular 600-cell, 3 <= n <= 120.

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