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Detour index of the n-triangular grid graph.
0

%I #14 Jun 24 2017 00:58:47

%S 0,6,69,399,1467,4197,10203,22047,43557,80187,139422,231228,368547,

%T 567837,849657,1239297,1767453,2470947,3393492,4586502,6109947,

%U 8033253,10436247,13410147,17058597,21498747,26862378,33297072,40967427,50056317,60766197

%N Detour index of the n-triangular grid graph.

%C 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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DetourIndex.html">Detour Index</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGridGraph.html">Triangular Grid Graph</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).

%F 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

%t 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}]

%t 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}]

%t Join[{0, 6, 69, 399}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {3, 72, 402, 1467, 4197, 10203, 22047}, {4, 20}]]

%t 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]

%K nonn

%O 0,2

%A _Eric W. Weisstein_, Jun 13 2017

%E a(8)-a(30) from _Andrew Howroyd_, Jun 19 2017