A124350 a(n) = 4*n*(floor(n^2/2)+1). For n >= 3, this is the number of directed Hamiltonian paths on the n-prism graph.

0, 4, 24, 60, 144, 260, 456, 700, 1056, 1476, 2040, 2684, 3504, 4420, 5544, 6780, 8256, 9860, 11736, 13756, 16080, 18564, 21384, 24380, 27744, 31300, 35256, 39420, 44016, 48836, 54120, 59644, 65664, 71940, 78744, 85820, 93456, 101380, 109896, 118716

Offset: 0

Author

Eric W. Weisstein, Oct 26 2006

Keywords

• nonn
• easy

Links

• Stefano Spezia, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Hamiltonian Path.
• Eric Weisstein's World of Mathematics, Prism Graph.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A080827, A124349.

Programs

Mathematica

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 4, 24, 60, 144, 260}, 60] (* Vincenzo Librandi, Jan 26 2016 *)

PARI

Vec(4*x*(x^2+1)*(x^2+4*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 06 2013

Formula

From Colin Barker, Sep 06 2013: (Start)

a(n) = n*(3 + (-1)^n + 2*n^2).

G.f.: 4*x*(x^2+1)*(x^2+4*x+1) / ((x-1)^4*(x+1)^2). (End)

a(n) = 4*n*A080827(n). - R. J. Mathar, Jan 25 2016

E.g.f.: 2*x*((2 + 3*x + x^2)*cosh(x) + (3 + 3*x + x^2)*sinh(x)). - Stefano Spezia, Jan 27 2024

Extensions

Formula and further terms from Max Alekseyev, Feb 07 2008