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A218489 The sequence of coefficients of cubic polynomials p(x+n), where p(x) = x^3 - 3*x + 1. 1

%I #22 May 17 2013 12:32:54

%S 1,0,-3,1,1,3,0,-1,1,6,9,3,1,9,24,19,1,12,45,53,1,15,72,111,1,18,105,

%T 199,1,21,144,323,1,24,189,489,1,27,240,703,1,30,297,971,1,33,360,

%U 1299,1,36,429,1693,1,39,504,2159,1,42,585,2703,1,45,672,3331

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

%C We note that p(x) = (x - s(1))*(x + c(1))*(x - c(2)),

%C p(x+1) = x^3 + 3*x^2 -1 = (x + s(1)*c(1))*(x - s(1)*c(2))*(x + c(1)*c(2)), p(x+2) = x^3 + 6*x^2 + 9*x + 3 = (x + c(1/2)^2)*(x + s(2)^2)*(x + s(4)^2), and p(x + n) = (x + n - 2 + c(1/2)^2)*(x + n - 2 + s(2)^2)*(x + n - 2 + s(4)^2), n = 2,3,..., 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 is the sequence of coefficients of polynomials p(x-n).

%F We have a(4*k) = 1, a(4*k + 1) = 3*k, a(4*k + 2) = 3*k^2 - 3, and a(4*k + 3) = k^3 - 3*k + 1. Moreover we obtain

%F b(k+1) = b(k) + 3, c(k+1) = 2*b(k) + c(k) + 3, d(k+1) = b(k) + c(k) + d(k) + 1, where p(x + k) = x^3 + b(k)*x^2 + c(k)*x + d(k).

%F Empirical g.f.: -(3*x^15-3*x^13+x^12-13*x^11+9*x^10+6*x^9-3*x^8+5*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 Cf. A214699, A214779, A215634, A215664, A215666, A215917, A215919.

%K sign

%O 0,3

%A _Roman Witula_, Oct 30 2012

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)