A054622 Number of ways to color vertices of an octagon using <= n colors, allowing only rotations.

0, 1, 36, 834, 8230, 48915, 210126, 720916, 2097684, 5381685, 12501280, 26796726, 53750346, 101969959, 184478490, 320367720, 536879176, 871980201, 1377508284, 2122961770, 3200020110, 4727881851, 6859513606, 9788908284, 13759455900

Offset

0,3

Comments

Length-8 necklaces with n kinds of beads. - Joerg Arndt, Apr 29 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = Sum_{d|8} phi(d)*n^(8/d)/8 = n*(n+1)*(n^6-n^5+n^4-n^3+2*n^2-2*n+4)/8.

G.f.: x*(1+27*x+546*x^2+1936*x^3+1971*x^4+525*x^5+34*x^6)/(1-x)^9. - Colin Barker, Jan 29 2012

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9). Vincenzo Librandi, Apr 29 2012

Mathematica

CoefficientList[Series[x*(1+27*x+546*x^2+1936*x^3+ 1971*x^4+525*x^5+34*x^6)/(1-x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Apr 29 2012 *)

Prog

(Magma) I:=[0, 1, 36, 834, 8230, 48915, 210126, 720916, 2097684]; [n le 9 select I[n] else 9*Self(n-1)-36*Self(n-2)+84*Self(n-3)-126*Self(n-4)+126*Self(n-5)-84*Self(n-6)+36*Self(n-7)-9*Self(n-8)+Self(n-9): n in [1..30]]; // Vincenzo Librandi, Apr 29 2012

Crossrefs

Row 8 of A075195.

Cf. A064621, A054623, ...

Sequence in context: A114238 A393010 A028163 * A028108 A001457 A203271

Adjacent sequences: A054619 A054620 A054621 * A054623 A054624 A054625

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 16 2000

Extensions

Edited by Christian G. Bower, Sep 07 2002

Status

approved