A288719 Detour index of the n-triangular grid graph.

0, 6, 69, 399, 1467, 4197, 10203, 22047, 43557, 80187, 139422, 231228, 368547, 567837, 849657, 1239297, 1767453, 2470947, 3393492, 4586502, 6109947, 8033253, 10436247, 13410147, 17058597, 21498747, 26862378, 33297072, 40967427, 50056317, 60766197

Offset

0,2

Comments

With at least 5 vertices per side, a Hamiltonian path exists between any two vertices except in the case of a pair of vertices adjacent to a corner when the longest path will include every vertex except the corner vertex. This leads to the formula v*(v-1)^2/2-3 where v is the number of vertices. - Andrew Howroyd, Jun 19 2017

Links

Table of n, a(n) for n=0..30.

Eric Weisstein's World of Mathematics, Detour Index

Eric Weisstein's World of Mathematics, Triangular Grid Graph

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Formula

a(n) = (n^6 + 9*n^5 + 29*n^4 + 39*n^3 + 18*n^2 - 48)/16 for n>3. - Andrew Howroyd, Jun 19 2017

Mathematica

Table[Piecewise[{{0, n == 0}, {6, n == 1}, {69, n == 2}, {399, n == 3}}, n^2 (n + 1) (n + 2) (n + 3)^2/16 - 3], {n, 0, 10}]
Table[Piecewise[{{0, n == 0}, {6, n == 1}, {69, n == 2}, {399, n == 3}}, 3 (Binomial[n + 3, 4] n (n + 3)/2 - 1)], {n, 0, 10}]
Join[{0, 6, 69, 399}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {3, 72, 402, 1467, 4197, 10203, 22047}, {4, 20}]]
CoefficientList[Series[-((3 x (2 + 9 x + 14 x^2 - 29 x^3 + 34 x^4 - 15 x^5 - 8 x^6 + 13 x^7 - 6 x^8 + x^9))/(-1 + x)^7), {x, 0, 20}], x]

Crossrefs

Sequence in context: A197170 A183438 A296016 * A201535 A198699 A346938

Adjacent sequences: A288716 A288717 A288718 * A288720 A288721 A288722

Keyword

nonn

Author

Eric W. Weisstein, Jun 13 2017

Extensions

a(8)-a(30) from Andrew Howroyd, Jun 19 2017

Status

approved