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

%I A060446 #19 Feb 16 2025 08:32:44
%S A060446 0,1,8,39,136,377,888,1855,3536,6273,10504,16775,25752,38233,55160,
%T A060446 77631,106912,144449,191880,251047,324008,413049,520696,649727,803184,
%U A060446 984385,1196936,1444743,1732024,2063321,2443512,2877823
%N A060446 Number of ways to color vertices of a pentagon using <= n colors, allowing rotations and reflections.
%C A060446 a(n) is also the number of 5-cycles in the (n+4)-path complement graph, - _Eric W. Weisstein_, Apr 11 2018
%H A060446 Harry J. Smith, <a href="/A060446/b060446.txt">Table of n, a(n) for n = 0..1000</a>
%H A060446 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A060446 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathComplementGraph.html">Path Complement Graph</a>
%H A060446 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A060446 a(n) = (n^5+5*n^3+4*n)/10.
%F A060446 G.f.: x*(1+2*x+6*x^2+2*x^3+x^4)/(1-x)^6. - _Colin Barker_, Jan 29 2012
%F A060446 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
%t A060446 Table[n (n^2 + 1) (n^2 + 4)/10, {n, 0, 20}] (* _Eric W. Weisstein_, Apr 11 2018 *)
%t A060446 LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 8, 39, 136, 377, 888}, {0, 20}] (* _Eric W. Weisstein_, Apr 11 2018 *)
%t A060446 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 *)
%o A060446 (PARI) for (n=0, 1000, write("b060446.txt", n, " ", (n^5 + 5*n^3 + 4*n)/10); ) \\ _Harry J. Smith_, Jul 05 2009
%Y A060446 Cf. A054620.
%Y A060446 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}).
%K A060446 nonn,easy
%O A060446 0,3
%A A060446 _N. J. A. Sloane_, Apr 07 2001