A108354 Expansion of 1/((1-x)^2(1+x^2)^2) in powers of x.

1, 2, 1, 0, 2, 4, 2, 0, 3, 6, 3, 0, 4, 8, 4, 0, 5, 10, 5, 0, 6, 12, 6, 0, 7, 14, 7, 0, 8, 16, 8, 0, 9, 18, 9, 0, 10, 20, 10, 0, 11, 22, 11, 0, 12, 24, 12, 0, 13, 26, 13, 0, 14, 28, 14, 0, 15, 30, 15, 0, 16, 32, 16, 0, 17, 34, 17, 0, 18, 36, 18, 0, 19, 38, 19, 0, 20, 40, 20, 0, 21, 42, 21, 0, 22

Offset

0,2

Comments

Self-convolution transform of A133872. - R. J. Mathar, Apr 06 2008

Links

Table of n, a(n) for n=0..84.

Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Formula

a(n)=2a(n-1)-3a(n-2)+4a(n-3)-3a(n-4)+2a(n-5)-a(n-6); a(n)=cos(pi*n/2)/4+(n+3)*sin(pi*n/2)/4+(n+3)/4.

Euler transform of length 4 sequence [ 2, -2, 0, 2]. - Michael Somos, Aug 17 2014

0 = a(n)*(+2*a(n+2) - a(n+3)) + a(n+1)*(-a(n+2) + 2*a(n+3)) for all n in Z. - Michael Somos, Aug 17 2014

Example

G.f. = 1 + 2*x + x^2 + 2*x^4 + 4*x^5 + 2*x^6 + 3*x^8 + 6*x^9 + 3*x^10 + ...

Mathematica

CoefficientList[Series[1/((1-x)^2 (1+x^2)^2), {x, 0, 100}], x] (* or *)
LinearRecurrence[{2, -3, 4, -3, 2, -1}, {1, 2, 1, 0, 2, 4}, 100](* Harvey P. Dale, Apr 11 2020 *)

Prog

(PARI) {a(n) = (n\4+1) * [1, 2, 1, 0][n%4+1]};

Crossrefs

Cf. A133872.

Sequence in context: A174026 A226075 A279226 * A329099 A329686 A261615

Adjacent sequences: A108351 A108352 A108353 * A108355 A108356 A108357

Keyword

easy,nonn

Author

Paul Barry, May 31 2005

Status

approved