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
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).
Programs
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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
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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 *)
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
Extensions
Edited by Christian G. Bower, Sep 07 2002
Comments