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

Author

N. J. A. Sloane, Apr 16 2000

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A054622. Row 7 of A075195.

Programs

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

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 *)

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

Extensions

Edited by Christian G. Bower, Sep 07 2002