A218489 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, -1, 1, 6, 9, 3, 1, 9, 24, 19, 1, 12, 45, 53, 1, 15, 72, 111, 1, 18, 105, 199, 1, 21, 144, 323, 1, 24, 189, 489, 1, 27, 240, 703, 1, 30, 297, 971, 1, 33, 360, 1299, 1, 36, 429, 1693, 1, 39, 504, 2159, 1, 42, 585, 2703, 1, 45, 672, 3331

Offset

0,3

Comments

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

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.

A218332 is the sequence of coefficients of polynomials p(x-n).

Links

Table of n, a(n) for n=0..63.

Formula

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

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).

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

Crossrefs

Cf. A214699, A214779, A215634, A215664, A215666, A215917, A215919.

Sequence in context: A362019 A016564 A104443 * A218332 A322506 A103496

Adjacent sequences: A218486 A218487 A218488 * A218490 A218491 A218492

Keyword

sign

Author

Roman Witula, Oct 30 2012

Status

approved