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
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jun 13 2017
EXTENSIONS
a(8)-a(30) from Andrew Howroyd, Jun 19 2017
STATUS
approved