A302695 Number of 6-cycles in the (n+5)-path complement graph.

0, 5, 50, 265, 996, 2985, 7610, 17185, 35320, 67341, 120770, 205865, 336220, 529425, 807786, 1199105, 1737520, 2464405, 3429330, 4691081, 6318740, 8392825, 11006490, 14266785, 18295976, 23232925, 29234530, 36477225, 45158540, 55498721, 67742410, 82160385, 99051360

Offset: 0

Author

Eric W. Weisstein, Apr 11 2018

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Path Complement Graph
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000292 (3-cycles of \bar P_{n+4}), A002817 (4-cycles of \bar P_{n+4}), A060446 (5-cycles of \bar P_{n+3}).

Programs

Mathematica

Table[n (4 + 22 n + 17 n^2 + 13 n^3 + 3 n^4 + n^5)/12, {n, 0, 20}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {5, 50, 265, 996, 2985, 7610, 17185}, {0, 20}]
CoefficientList[Series[x (-5 - 15 x - 20 x^2 - 16 x^3 - 3 x^4 - x^5)/(-1 + x)^7, {x, 0, 20}], x]

PARI

a(n) = n*(4+22*n+17*n^2+13*n^3+3*n^4+n^5)/12; \\ Altug Alkan, Apr 12 2018

Formula

G.f.: x*(-5 - 15*x - 20*x^2 - 16*x^3 - 3*x^4 - x^5)/(-1 + x)^7.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

a(n) = n*(4 + 22*n + 17*n^2 + 13*n^3 + 3*n^4 + n^5)/12.