A060446 Number of ways to color vertices of a pentagon using <= n colors, allowing rotations and reflections.

0, 1, 8, 39, 136, 377, 888, 1855, 3536, 6273, 10504, 16775, 25752, 38233, 55160, 77631, 106912, 144449, 191880, 251047, 324008, 413049, 520696, 649727, 803184, 984385, 1196936, 1444743, 1732024, 2063321, 2443512, 2877823

Offset: 0

Author

N. J. A. Sloane, Apr 07 2001

Keywords

• nonn
• easy

Comments

a(n) is also the number of 5-cycles in the (n+4)-path complement graph, - Eric W. Weisstein, Apr 11 2018

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• 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 (6,-15,20,-15,6,-1).

Crossrefs

Cf. A054620.

Cf. A000292 (3-cycle count of \bar P_{n+4}), A002817 (4-cycle count of \bar P_{n+4}), A302695 (6-cycle count of \bar P_{n+5}).

Programs

Mathematica

Table[n (n^2 + 1) (n^2 + 4)/10, {n, 0, 20}] (* Eric W. Weisstein, Apr 11 2018 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 8, 39, 136, 377, 888}, {0, 20}] (* Eric W. Weisstein, Apr 11 2018 *)
CoefficientList[Series[x (1 + 2 x + 6 x^2 + 2 x^3 + x^4)/(-1 + x)^6, {x, 0, 20}], x] (* Eric W. Weisstein, Apr 11 2018 *)

PARI

for (n=0, 1000, write("b060446.txt", n, " ", (n^5 + 5*n^3 + 4*n)/10); ) \\ Harry J. Smith, Jul 05 2009

Formula

a(n) = (n^5+5*n^3+4*n)/10.

G.f.: x*(1+2*x+6*x^2+2*x^3+x^4)/(1-x)^6. - Colin Barker, Jan 29 2012

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Eric W. Weisstein, Apr 11 2018