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A145401
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Number of Hamiltonian cycles in P_6 X P_n.
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4
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0, 1, 4, 37, 154, 1072, 5320, 32675, 175294, 1024028, 5668692, 32463802, 181971848, 1033917350, 5824476298, 32989068162, 186210666468, 1053349394128, 5950467515104, 33643541208290, 190115484271760, 1074685815276400, 6073680777522430, 34330607094625734
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OFFSET
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1,3
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REFERENCES
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F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (5,14,-63,12,90,-35,-66,118,-8,-82,42,28,-4,2).
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FORMULA
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Recurrence: a(1) = 0,
a(2) = 1,
a(3) = 4,
a(4) = 37,
a(5) = 154,
a(6) = 1072,
a(7) = 5320,
a(8) = 32675,
a(9) = 175294,
a(10) = 1024028,
a(11) = 5668692,
a(12) = 32463802,
a(13) = 181971848,
a(14) = 1033917350, and
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).
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]
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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