A100888 Expansion of (3+x-x^2)/((1+x+x^2)(1-x-x^2)).

3, 1, 2, 7, 7, 12, 23, 33, 54, 91, 143, 232, 379, 609, 986, 1599, 2583, 4180, 6767, 10945, 17710, 28659, 46367, 75024, 121395, 196417, 317810, 514231, 832039, 1346268, 2178311, 3524577, 5702886, 9227467, 14930351, 24157816, 39088171, 63245985

Offset: 0

Author

Creighton Dement, Nov 21 2004

Keywords

• nonn
• easy

Comments

This sequence was investigated in cooperation with Paul Barry. Generating floretion: - 0.5'i - 0.5'k - 0.5j' - 0.5'ii' + 0.5'jj' - 0.5'kk' + 0.5'ik' - 0.5'ki' ("jes"). A100885(n) = (1/2)(A100886(n) + A100887(n) - a(n)).

Crossrefs

Cf. A100885, A100886, A100887, A100889, A100890.

Programs

Mathematica

a[0] = 3; a[1] = 1; a[2] = 2; a[3] = 7; a[n_] := a[n] = a[n - 2] + 2a[n - 3] + a[n - 4]; Table[ a[n], {n, 0, 37}] (* Robert G. Wilson v, Nov 26 2004 *)
CoefficientList[ Series[(3 + x - x^2)/((1 + x + x^2)(1 - x - x^2)), {x, 0, 37}], x] (* Robert G. Wilson v, Nov 26 2004 *)

PARI

Vec((3+x-x^2)/((1+x+x^2)*(1-x-x^2))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = Fib(n+2) + sqrt(3)cos(2Pi*n/3 + Pi/6) + sin(2Pi*n/3 + Pi/6); a(n) = a(n-2) + 2a(n-3) + a(n-4), a(0) = 3, a(1) = 1, a(2) = 2, a(3) = 7.

Extensions

More terms from Robert G. Wilson v, Nov 26 2004