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A092181
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Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).
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9
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1, 24, 153, 544, 1425, 3096, 5929, 10368, 16929, 26200, 38841, 55584, 77233, 104664, 138825, 180736, 231489, 292248, 364249, 448800, 547281, 661144, 791913, 941184, 1110625, 1301976, 1517049, 1757728, 2025969, 2323800, 2653321, 3016704
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OFFSET
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1,2
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COMMENTS
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This is the 4-dimensional regular convex polytope called the 24-cell, hyperdiamond or icositetrachoron.
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LINKS
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Eric Weisstein's World of Mathematics, 24-Cell
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FORMULA
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a(n) = n^2*((3*n^2)-(4*n)+2).
a(n) = C(n+3,4) + 19 C(n+2,4) + 43 C(n+1,4) + 9 C(n,4).
a(n) = +5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5). G.f.: x*(1+19*x+43*x^2+9*x^3)/(1-x)^5. [R. J. Mathar, Jun 21 2010]
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EXAMPLE
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a(3)= 3^2*((3*3^2)-(4*3)+2) = 9*(27-12+2) = 9*17 = 153
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MATHEMATICA
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Table[SeriesCoefficient[x (1 + 19 x + 43 x^2 + 9 x^3)/(1 - x)^5, {x, 0, n}], {n, 32}] (* Michael De Vlieger, Dec 14 2015 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 24, 153, 544, 1425}, 40] (* Harvey P. Dale, May 25 2022 *)
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PROG
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Michael J. Welch (mjw1(AT)ntlworld.com), Mar 31 2004
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STATUS
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approved
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