A222199
Number of Hamiltonian cycles in the graph C_n X C_n.
Original entry on oeis.org
48, 1344, 23580, 3273360, 257165468, 171785923808, 61997157648756, 196554899918485160
Offset: 3
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Table[Length[FindHamiltonianCycle[GraphProduct[CycleGraph[n], CycleGraph[n], "Cartesian"], All]], {n, 3, 6}] (* Eric W. Weisstein, Dec 16 2023 *)
A278417
a(n) = n*((2+sqrt(3))^n + (2-sqrt(3))^n)/2.
Original entry on oeis.org
0, 2, 14, 78, 388, 1810, 8106, 35294, 150536, 632034, 2620870, 10759342, 43804812, 177105266, 711809378, 2846259390, 11330543632, 44929049794, 177540878718, 699402223118, 2747583822740, 10766828545746, 42095796462874, 164244726238366, 639620518118424, 2486558615814050, 9651161613824822, 37403957244654702
Offset: 0
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f:=n->expand(n*((2+sqrt(3))^n + (2-sqrt(3))^n)/2); # N. J. A. Sloane, May 13 2017
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Table[Simplify[(n/2) (((2 + #)^n + (2 - #)^n)) &@ Sqrt@ 3], {n, 3, 27}] (* or *)
Drop[#, 3] &@ CoefficientList[Series[2 x^3*(39 - 118 x + 55 x^2 - 7 x^3)/(1 - 4 x + x^2)^2, {x, 0, 27}], x] (* Michael De Vlieger, Nov 24 2016 *)
LinearRecurrence[{8,-18,8,-1},{0,2,14,78},30] (* Harvey P. Dale, Jan 01 2021 *)
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vector(25, n, n+=2; n*((2+sqrt(3))^n + ((2-sqrt(3))^n))/2) \\ Colin Barker, Nov 21 2016
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Vec(2*x^3*(39 - 118*x + 55*x^2 - 7*x^3) / (1 - 4*x + x^2)^2 + O(x^30)) \\ Colin Barker, Nov 21 2016
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def a278417(n):
a = [0, 2, 14, 78, 388, 1810]
if n < 6:
return a[n]
for k in range(n - 5):
a = a[1:] + [7*a[-1] - 10*a[-2] - 10*a[-3] + 7*a[-4] - a[-5]]
return a[-1]
# David Radcliffe, May 09 2025
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