cp's OEIS Frontend

A218332 The sequence of coefficients of cubic polynomials p(x-n), where p(x) = x^3 - 3*x + 1.

%I A218332 #27 May 17 2013 12:33:03
%S A218332 1,0,-3,1,1,-3,0,3,1,-6,9,-1,1,-9,24,-17,1,-12,45,-51,1,-15,72,-109,1,
%T A218332 -18,105,-197,1,-21,144,-321,1,-24,189,-487,1,-27,240,-701,1,-30,297,
%U A218332 -969,1,-33,360,-1297,1,-36,429,-1691,1,-39,504,-2157,1,-42,585,-2701
%N A218332 The sequence of coefficients of cubic polynomials p(x-n), where p(x) = x^3 - 3*x + 1.
%C A218332 We note that p(x) =  (x - s(1))*(x + c(1))*(x - c(2)),
%C A218332 p(x-1) = x^3 - 3*x^2 + 3 = (x - s(2)*c(1/2))*(x - s(4)*c(1/2))*(x + s(2)*s(4)), p(x-2) = x^3 - 6*x^2 + 9*x - 1 = (x - c(1)^2)*(x - c(2)^2)*(x - c(4)^2), and p(x - n - 2) = (x - n - c(1)^2)*(x - n -c(2)^2)*(x - n - c(4)^2), n = 1,2,..., where c(j) := 2*cos(Pi*j/9) and s(j) = 2*sin(Pi*j/18). These one's are characteristic polynomials many sequences A... - see crossrefs.
%C A218332 A218489 is the sequence of coefficients of polynomials p(x+n).
%F A218332 We have a(4*k) = 1, a(4*k+1) = -3*k, a(4*k+2) = 3*k^2 - 3, a(4*k+3) = -k^3 + 3*k + 1. Moreover we obtain the relations b(k+1) = b(k) - 3, c(k+1) = c(k) - 2*b(k) + 3, b(k) - c(k) + d(k) - 1, whenever p(x-k) = x^3 + b(k)*x^2 + c(k)*x + d(k).
%F A218332 Empirical g.f.: (x^15-3*x^13-x^12-7*x^11-9*x^10+6*x^9+3*x^8-x^7+12*x^6-3*x^5-3*x^4+x^3-3*x^2+1) / ((x-1)^4*(x+1)^4*(x^2+1)^4). - _Colin Barker_, May 17 2013
%Y A218332 Cf. A214699, A214779, A215455, A215634, A215635, A215664, A215665.
%K A218332 sign
%O A218332 0,3
%A A218332 _Roman Witula_, Nov 02 2012