A054621 Number of ways to color vertices of a heptagon using <= n colors, allowing only rotations.

0, 1, 20, 315, 2344, 11165, 39996, 117655, 299600, 683289, 1428580, 2783891, 5118840, 8964085, 15059084, 24408495, 38347936, 58619825, 87460020, 127695979, 182857160, 257298381, 356336860, 486403655, 655210224, 871930825, 1147401476, 1494336195, 1927561240

Offset

0,3

Comments

Length-7 necklaces with n kinds of beads. - Vincenzo Librandi Apr 30 2012

Links

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

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Formula

a(n) = (1/7) * Sum_{d|7} phi(d)*n^(7/d) = (1/7) * (n^7 + 6*n). [corrected by Klaus Wagner, Sep 15 2015]

G.f.: x*(1+12*x+183*x^2+328*x^3+183*x^4+12*x^5+x^6) / (1-x)^8. - R. J. Mathar, Aug 30 2011

a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8), n>7. - Wesley Ivan Hurt, Sep 15 2015

Maple

a:=proc(n) option remember:
if n=0 then 0 elif n=1 then 1 elif n=2 then 20 elif n=3 then 315 elif n=4 then 2344 elif n=5 then 11165 elif n=6 then 39996 elif n=7 then 117655 elif n=8 then 299600 else
8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8): fi: end: seq(a(n), n=0..50); # Wesley Ivan Hurt, Sep 15 2015

Mathematica

CoefficientList[Series[x*(1+12*x+183*x^2+328*x^3+183*x^4+ 12*x^5+x^6)/(x-1)^8, {x, 0, 33}], x] (* Vincenzo Librandi, Apr 30 2012 *)

Prog

(Magma) I:=[0, 1, 20, 315, 2344, 11165, 39996, 117655]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5)-28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..30]]; // Vincenzo Librandi, Apr 30 2012

Crossrefs

Cf. A054622. Row 7 of A075195.

Sequence in context: A281931 A034094 A011197 * A111778 A024387 A307173

Adjacent sequences: A054618 A054619 A054620 * A054622 A054623 A054624

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 16 2000

Extensions

Edited by Christian G. Bower, Sep 07 2002

Status

approved