A296779 Detour index of the n X n grid graph.

0, 16, 256, 1744, 6912, 21744, 55296, 126016, 256000, 487600, 864000, 1467216, 2370816, 3716944, 5619712, 8306944, 11943936, 16875216, 23328000, 31800400, 42592000, 56397616, 73598976, 95137344, 121485312, 153887344, 192914176, 240177616, 296352000, 363488400

Offset

1,2

Comments

Also, the maximum detour index of any bipartite graph on n^2 nodes. - Andrew Howroyd, Dec 20 2017

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Detour Index

Eric Weisstein's World of Mathematics, 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(2n-1) = 32*n^3*(n-1)^3, a(2n) = 4*n^2*(8*n^4 - 5*n^2 + 1). - Andrew Howroyd, Dec 20 2017

From Colin Barker, Dec 21 2017: (Start)

G.f.: 16*x^2*(1 + 14*x + 73*x^2 + 160*x^3 + 224*x^4 + 160*x^5 + 73*x^6 + 14*x^7 + x^8) / ((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>12.

(End)

a(n) = A296819(n^2). - Andrew Howroyd, Dec 23 2017

Mathematica

a[n_] := If[OddQ[n], (1/2)*(n-1)^3*(n+1)^3, (1/4)*n^2*(2*n^4 - 5*n^2 + 4)]; Array[a, 30] (* Jean-François Alcover, Dec 21 2017, after Andrew Howroyd *)
LinearRecurrence[{2, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1}, {0, 16, 256, 1744, 6912, 21744, 55296, 126016, 256000, 487600, 864000, 1467216}, 30] (* Eric W. Weisstein, Dec 21 2017 *)
CoefficientList[Series[16 x (1 + 14 x + 73 x^2 + 160 x^3 + 224 x^4 + 160 x^5 + 73 x^6 + 14 x^7 + x^8)/((1 - x)^7 (1 + x)^5), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 21 2017 *)

Prog

(PARI) a(n)=((n^2)*(n^2-1)^2 - if(n%2, (n+1)^2*(n-1)^2, n^2*(n^2/2-1)))/2; \\ Andrew Howroyd, Dec 20 2017
(PARI) concat(0, Vec(16*x^2*(1 + 14*x + 73*x^2 + 160*x^3 + 224*x^4 + 160*x^5 + 73*x^6 + 14*x^7 + x^8) / ((1 - x)^7*(1 + x)^5) + O(x^40))) \\ Colin Barker, Dec 21 2017

Crossrefs

Cf. A296780, A296781, A296784, A296819.

Sequence in context: A189619 A189691 A188742 * A188994 A208075 A223641

Adjacent sequences: A296776 A296777 A296778 * A296780 A296781 A296782

Keyword

nonn,easy

Author

Eric W. Weisstein, Dec 20 2017

Extensions

Terms a(8) and beyond from Andrew Howroyd, Dec 20 2017

Status

approved