A296780 Detour index of the n X n X n grid graph.

0, 184, 8788, 126016, 953312, 4980744, 20000844, 66781696, 192914176, 498751000, 1176318500, 2576159424, 5295012768, 10321114216, 19204598812, 34338770944, 59257739264, 99137135736, 161273290356, 255920008000, 397011388000, 603492894664, 900354295148

Offset

1,2

Comments

Also, the maximum detour index of any bipartite graph on n^3 nodes. - Andrew Howroyd, Dec 21 2017

Links

Colin Barker, 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 (3,4,-20,0,56,-28,-84,70,70,-84,-28,56,0,-20,4,3,-1).

Formula

a(2n-1) = 4*(n - 1)^3*(4*n^2 - 2*n + 1)^3, a(2n) = 8*n^3*(32*n^6 - 10*n^3 + 1). - Andrew Howroyd, Dec 21 2017

From Colin Barker, Dec 21 2017: (Start)

G.f.: 4*x^2*(46 + 2059*x + 24729*x^2 + 135948*x^3 + 448126*x^4 + 938845*x^5 + 1358863*x^6 + 1346304*x^7 + 941714*x^8 + 442773*x^9 + 138599*x^10 + 25508*x^11 + 2482*x^12 + 83*x^13 + x^14) / ((1 - x)^10*(1 + x)^7).

a(n) = 3*a(n-1) + 4*a(n-2) - 20*a(n-3) + 56*a(n-5) - 28*a(n-6) - 84*a(n-7) + 70*a(n-8) + 70*a(n-9) - 84*a(n-10) - 28*a(n-11) + 56*a(n-12) - 20*a(n-14) + 4*a(n-15) + 3*a(n-16) - a(n-17) for n>17.

(End)

a(n) = A296819(n^3). - Andrew Howroyd, Dec 23 2017

Mathematica

Table[Piecewise[{{n^3 (n^6 - 5 n^3/2 + 2)/2, Mod[n, 2] == 0}, {(n - 1)^3 (n^2 + n + 1)^3/2, Mod[n, 2] == 1}}], {n, 20}]
LinearRecurrence[{3, 4, -20, 0, 56, -28, -84, 70, 70, -84, -28, 56, 0, -20, 4, 3, -1}, {0, 184, 8788, 126016, 953312, 4980744, 20000844, 66781696, 192914176, 498751000, 1176318500, 2576159424, 5295012768, 10321114216, 19204598812, 34338770944, 59257739264}, 20]
CoefficientList[Series[4 x (46 + 2059 x + 24729 x^2 + 135948 x^3 + 448126 x^4 + 938845 x^5 + 1358863 x^6 + 1346304 x^7 + 941714 x^8 + 442773 x^9 + 138599 x^10 + 25508 x^11 + 2482 x^12 + 83 x^13 + x^14)/((1 - x)^10 (1 + x)^7), {x, 0, 20}], x]

Prog

(PARI) a(n) = if(n%2, (n-1)^3*(n^2 + n + 1)^3, n^3*(n^6 - 5*n^3/2 + 2))/2; \\ Andrew Howroyd, Dec 21 2017
(PARI) concat(0, Vec(4*x^2*(46 + 2059*x + 24729*x^2 + 135948*x^3 + 448126*x^4 + 938845*x^5 + 1358863*x^6 + 1346304*x^7 + 941714*x^8 + 442773*x^9 + 138599*x^10 + 25508*x^11 + 2482*x^12 + 83*x^13 + x^14) / ((1 - x)^10*(1 + x)^7) + O(x^40))) \\ Colin Barker, Dec 21 2017

Crossrefs

Cf. A296779, A296819.

Sequence in context: A221931 A223117 A221782 * A035831 A094631 A060491

Adjacent sequences: A296777 A296778 A296779 * A296781 A296782 A296783

Keyword

nonn,easy

Author

Eric W. Weisstein, Dec 20 2017

Extensions

Terms a(4) and beyond from Andrew Howroyd, Dec 21 2017

Status

approved