A134172 Expansion of x^2*(1+x)*(1-x+x^2) / ((1-x)^2*(1+x^2)^2).

0, 0, 1, 2, 1, 1, 4, 5, 2, 2, 7, 8, 3, 3, 10, 11, 4, 4, 13, 14, 5, 5, 16, 17, 6, 6, 19, 20, 7, 7, 22, 23, 8, 8, 25, 26, 9, 9, 28, 29, 10, 10, 31, 32, 11, 11, 34, 35, 12, 12, 37, 38, 13, 13, 40, 41, 14, 14, 43, 44, 15, 15, 46, 47, 16, 16, 49, 50, 17, 17, 52, 53, 18, 18, 55, 56, 19, 19, 58

Offset: 0

Author

Paul Curtz, Jan 13 2008

Keywords

• nonn
• easy

Comments

Old definition was: "Starting with 0, 1, 2, 3, ... (A001477), write 0, 0 instead of a(0), 1, 1 instead of a(3) and in general n, n instead of a(3n)".

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Programs

Mathematica

LinearRecurrence[{2, -3, 4, -3, 2, -1}, {0, 0, 1, 2, 1, 1}, 50] (* G. C. Greubel, May 29 2016 *)

PARI

concat(vector(2), Vec(x^2*(1+x)*(1-x+x^2)/((1-x)^2*(1+x^2)^2) + O(x^50))) \\ Colin Barker, May 30 2016

Formula

From Colin Barker, May 30 2016: (Start)

G.f.: x^2*(1+x)*(1-x+x^2) / ((1-x)^2*(1+x^2)^2).

a(n) = (-2+(-i)^n+i^n+(4-(1+i)*(-i)^n-(1-i)*i^n)*n)/8 where i = sqrt(-1).

a(n) = 2*a(n-1)-3*a(n-2)+4*a(n-3)-3*a(n-4)+2*a(n-5)-a(n-6) for n>5. (End)

Extensions

New name using the g.f. from Colin Barker, May 30 2016