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 A288719 Detour index of the n-triangular grid graph. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified September 19 17:19 EDT 2021. Contains 347564 sequences. (Running on oeis4.)