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A296784 Detour index for the n X n torus grid graph. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified September 30 02:05 EDT 2022. Contains 357095 sequences. (Running on oeis4.)