A296784 Detour index for the n X n torus grid graph.

288, 1744, 7200, 21744, 56448, 126016, 259200, 487600, 871200, 1467216, 2384928, 3716944, 5644800, 8306944, 11985408, 16875216, 23392800, 31800400, 42688800, 56397616, 73738368, 95137344, 121680000, 153887344, 193179168, 240177616, 296704800, 363488400

Offset

3,1

Comments

The n X n torus grid graph is Hamilton-connected for odd n, giving a(n) = n^2*(n^2 - 1)^2/2 for odd n.

Links

Colin Barker, Table of n, a(n) for n = 3..1000

Eric Weisstein's World of Mathematics, Detour Index

Eric Weisstein's World of Mathematics, Torus Grid Graph

Index entries for linear recurrences with constant coefficients, signature (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Formula

a(n) = n^2*(n^2 - 1)^2/2 for odd n.

a(n) = A296779(n) = n^2*(2*n^4 - 5*n^2 + 4)/4 for even n. - Andrew Howroyd, Dec 21 2017

From Colin Barker, Dec 21 2017: (Start)

G.f.: 16*x^3*(18 + 73*x + 160*x^2 + 203*x^3 + 190*x^4 + 69*x^5 - 4*x^6 + 6*x^7 + 10*x^8 - 4*x^9 - 2*x^10 + x^11) / ((1 - x)^7*(1 + x)^5).

a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n>14.

(End)

Mathematica

a[n_] := If[OddQ[n], (1/2)*n^2*(n^2 - 1)^2, (1/4)*n^2*(2*n^4 - 5*n^2 + 4)]; Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Dec 21 2017, after Andrew Howroyd *)
LinearRecurrence[{2, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1}, {288, 1744, 7200, 21744, 56448, 126016, 259200, 487600, 871200, 1467216, 2384928, 3716944}, 20] (* Eric W. Weisstein, Dec 21 2017 *)
CoefficientList[Series[(16 (18 + 73 x + 160 x^2 + 203 x^3 + 190 x^4 + 69 x^5 - 4 x^6 + 6 x^7 + 10 x^8 - 4 x^9 - 2 x^10 + x^11))/((1 - x)^7 (1 + x)^5), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 21 2017 *)

Prog

(PARI) a(n) = n^2 * if(n%2, (n^2 - 1)^2/2, (2*n^4 - 5*n^2 + 4)/4); \\ Andrew Howroyd, Dec 21 2017
(PARI) Vec(16*x^3*(18 + 73*x + 160*x^2 + 203*x^3 + 190*x^4 + 69*x^5 - 4*x^6 + 6*x^7 + 10*x^8 - 4*x^9 - 2*x^10 + x^11) / ((1 - x)^7*(1 + x)^5) + O(x^40)) \\ Colin Barker, Dec 21 2017

Crossrefs

Cf. A296779.

Sequence in context: A280936 A250871 A128392 * A235078 A235072 A235769

Adjacent sequences: A296781 A296782 A296783 * A296785 A296786 A296787

Keyword

nonn,easy

Author

Eric W. Weisstein, Dec 20 2017

Extensions

Terms a(8) and beyond from Andrew Howroyd, Dec 21 2017

Status

approved