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 A167983 Number of n-cycles on the graph of the regular 24-cell, 3 <= n <= 24. 4

%I

%S 96,360,1440,7120,37728,196488,974592,4536000,19934208,82689264,

%T 322437312,1171745280,3924079104,11964375936,32761139328,79244294016,

%U 165800420352,291640320576,413774810112,443415854592,318534709248,114869295744

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

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

%H Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI/GP scripts for various math problems</a>

%H Max A. Alekseyev, GĂ©rard P. Michon, <a href="http://arxiv.org/abs/1602.01396">Making Walks Count: From Silent Circles to Hamiltonian Cycles</a>, arXiv:1602.01396 [math.CO], 2016.

%H A. Weimholt, <a href="http://www.weimholt.com/andrew/24.html">24-cell net</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/24-Cell.html">24-Cell </a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclePolynomial.html">Cycle Polynomial</a>

%e a(3) = 96, because there are 96 3-cycles on the graph of the 24-cell.

%e 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.

%Y Cf. A167981 - 2n-cycles on graph of the tesseract.

%Y Cf. A167982 - n-cycles on graph of 16-cell.

%Y Cf. A167984 - n-cycles on graph of 120-cell.

%Y Cf. A167985 - n-cycles on graph of 600-cell.

%Y Cf. A085452 - T(n, k) = 2k-cycles on graph of n-cube.

%Y Cf. A144151 - ignoring first three columns (0<=k<=2), T(n, k) gives k-cycles on (n-1)-simplex.

%Y Cf. A167986 - T(n, k) = k-cycles on graph of n-orthoplex.

%K fini,full,nonn

%O 3,1

%A _Andrew Weimholt_, Nov 16 2009

%E a(16)-a(24) and "full" keyword from _Max Alekseyev_, Nov 18 2009

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