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

0,3

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).

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

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 *)

Prog

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

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}).

Sequence in context: A200157 A101374 A163687 * A318267 A063002 A055581

Adjacent sequences: A060443 A060444 A060445 * A060447 A060448 A060449

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 07 2001

Status

approved