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%I #36 Jan 30 2024 08:22:44
%S 0,4,24,60,144,260,456,700,1056,1476,2040,2684,3504,4420,5544,6780,
%T 8256,9860,11736,13756,16080,18564,21384,24380,27744,31300,35256,
%U 39420,44016,48836,54120,59644,65664,71940,78744,85820,93456,101380,109896,118716
%N 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.
%H Stefano Spezia, <a href="/A124350/b124350.txt">Table of n, a(n) for n = 0..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrismGraph.html">Prism Graph</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F From _Colin Barker_, Sep 06 2013: (Start)
%F a(n) = n*(3 + (-1)^n + 2*n^2).
%F G.f.: 4*x*(x^2+1)*(x^2+4*x+1) / ((x-1)^4*(x+1)^2). (End)
%F a(n) = 4*n*A080827(n). - _R. J. Mathar_, Jan 25 2016
%F E.g.f.: 2*x*((2 + 3*x + x^2)*cosh(x) + (3 + 3*x + x^2)*sinh(x)). - _Stefano Spezia_, Jan 27 2024
%t LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 4, 24, 60, 144, 260}, 60] (* _Vincenzo Librandi_, Jan 26 2016 *)
%o (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
%Y Cf. A080827, A124349.
%K nonn,easy
%O 0,2
%A _Eric W. Weisstein_, Oct 26 2006
%E Formula and further terms from _Max Alekseyev_, Feb 07 2008
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