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%I #25 Jan 01 2019 06:31:05
%S 0,1,4,37,154,1072,5320,32675,175294,1024028,5668692,32463802,
%T 181971848,1033917350,5824476298,32989068162,186210666468,
%U 1053349394128,5950467515104,33643541208290,190115484271760,1074685815276400,6073680777522430,34330607094625734
%N Number of Hamiltonian cycles in P_6 X P_n.
%D F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
%H Alois P. Heinz, <a href="/A145401/b145401.txt">Table of n, a(n) for n = 1..400</a>
%H F. Faase, <a href="http://www.iwriteiam.nl/Cpaper.zip">On the number of specific spanning subgraphs of the graphs G X P_n</a>, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
%H F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>.
%H F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from the counting program</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (5,14,-63,12,90,-35,-66,118,-8,-82,42,28,-4,2).
%F Recurrence: a(1) = 0,
%F a(2) = 1,
%F a(3) = 4,
%F a(4) = 37,
%F a(5) = 154,
%F a(6) = 1072,
%F a(7) = 5320,
%F a(8) = 32675,
%F a(9) = 175294,
%F a(10) = 1024028,
%F a(11) = 5668692,
%F a(12) = 32463802,
%F a(13) = 181971848,
%F a(14) = 1033917350, and
%F a(n) = 5a(n-1) +14a(n-2) -63a(n-3) +12a(n-4) +90a(n-5) -35a(n-6) -66a(n-7) +118a(n-8) -8a(n-9) -82a(n-10) +42a(n-11) +28a(n-12) -4a(n-13) +2a(n-14).
%F G.f.: -x^2*(x -1)*(x^11 -x^10 +3*x^9 +12*x^8 -3*x^7 -3*x^4 +21*x^3 -3*x^2 -1)/(2*x^14 -4*x^13 +28*x^12 +42*x^11 -82*x^10 -8*x^9 +118*x^8 -66*x^7- 35*x^6 +90*x^5 +12*x^4 -63*x^3 +14*x^2 +5*x -1). [_Colin Barker_, Aug 31 2012]
%p a:= n-> (Matrix([32675, 5320, 1072, 154, 37, 4, 1, 0, 1/2, 0, -5, -16, 51, 869/2]). Matrix(14, (i, j)-> if i=j-1 then 1 elif j=1 then [5, 14, -63, 12, 90, -35, -66, 118, -8, -82, 42, 28, -4, 2][i] else 0 fi)^n)[1,9]: seq(a(n), n=1..30); # _Alois P. Heinz_, Oct 24 2009
%t a[n_] := ({32675, 5320, 1072, 154, 37, 4, 1, 0, 1/2, 0, -5, -16, 51, 869/2 }.MatrixPower[Table[If[i == j-1, 1, If[j == 1, {5, 14, -63, 12, 90, -35, -66, 118, -8, -82, 42, 28, -4, 2}[[i]], 0]], {i, 1, 14}, {j, 1, 14}], n] )[[9]]; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Feb 14 2016, after _Alois P. Heinz_ *)
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, Feb 03 2009
%E More terms from _Alois P. Heinz_, Oct 24 2009
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