A092181 Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).

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

Offset: 1

Author

Michael J. Welch (mjw1(AT)ntlworld.com), Mar 31 2004

Keywords

• easy
• nonn

Comments

This is the 4-dimensional regular convex polytope called the 24-cell, hyperdiamond or icositetrachoron.

Examples

			a(3)= 3^2*((3*3^2)-(4*3)+2) = 9*(27-12+2) = 9*17 = 153
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
• Eric Weisstein's World of Mathematics, 24-Cell
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). [_R. J. Mathar_, Jun 21 2010]

Crossrefs

Cf. A000332, A000583, A014820, A092182, A092183.

Programs

Magma

[n^2*((3*n^2)-(4*n)+2): n in [1..40]]; // Vincenzo Librandi, May 22 2011

Mathematica

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 *)

PARI

a(n) = n^2*(3*n^2-4*n+2); \\ Michel Marcus, Dec 14 2015

Formula

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]

a(n) = Sum_{k = 1..n} (k^3 + k^7)* binomial(n,k)/binomial(n+k,k). Cf. A034262 and A155977. - Peter Bala, Feb 12 2019