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

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, -18, 105, -197, 1, -21, 144, -321, 1, -24, 189, -487, 1, -27, 240, -701, 1, -30, 297, -969, 1, -33, 360, -1297, 1, -36, 429, -1691, 1, -39, 504, -2157, 1, -42, 585, -2701

Offset: 0

Author

Roman Witula, Nov 02 2012

Keywords

• sign

Comments

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

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.

A218489 is the sequence of coefficients of polynomials p(x+n).

Crossrefs

Cf. A214699, A214779, A215455, A215634, A215635, A215664, A215665.

Formula

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).

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