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A014524 Number of Hamiltonian paths from NW to SW corners in a grid with 2n rows and 4 columns.

%I A014524 #31 Jul 08 2025 05:40:43
%S A014524 0,1,8,47,264,1480,8305,46616,261663,1468752,8244304,46276385,
%T A014524 259755560,1458042831,8184190168,45938958232,257861540369,
%U A014524 1447411446840,8124514782015,45603992276896,255981331487648
%N A014524 Number of Hamiltonian paths from NW to SW corners in a grid with 2n rows and 4 columns.
%H A014524 Vincenzo Librandi, <a href="/A014524/b014524.txt">Table of n, a(n) for n = 0..1000</a>
%H A014524 K. L. Collins and L. B. Krompart, <a href="http://dx.doi.org/10.1016/0012-365X(95)00330-Y">The number of Hamiltonian paths in a rectangular grid</a>, Discrete Math. 169 (1997), 29-38.
%H A014524 <a href="/index/Gra#graphs">Index entries for sequences related to graphs, Hamiltonian</a>
%H A014524 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-9,7,-1).
%F A014524 From _Colin Barker_, May 20 2013: (Start)
%F A014524 a(n) = 7*a(n-1)-9*a(n-2)+7*a(n-3)-a(n-4).
%F A014524 G.f.: x*(x+1)/(x^4-7*x^3+9*x^2-7*x+1). (End)
%e A014524 Illustration of a(1)=1:
%e A014524    .__.__.__.
%e A014524    .__.__.__|
%e A014524 Illustration of a few of the 8 solutions to a(2):
%e A014524    .__.__.__.    .  .__.__.    .  .__.__.    .__.__.__.
%e A014524    .__.__.  |    |  |  .__|    |__|  .__|    .__.__.__|
%e A014524    |__   |  |    |__|  |__.    .__.  |__.    |__.__.__.
%e A014524    .__|  |__|    .__.__.__|    |  |__.__|    .__.__.__|
%t A014524 CoefficientList[Series[x (x + 1)/(x^4 - 7 x^3 + 9 x^2 - 7 x + 1), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 15 2013 *)
%Y A014524 Even bisection of column 4 of A271592.
%Y A014524 Cf. A000532, A181688, A014523, A014585, A003695, A006864.
%K A014524 nonn,easy
%O A014524 0,3
%A A014524 _N. J. A. Sloane_
%E A014524 Name clarified by _Andrew Howroyd_, Apr 10 2016