A115451 Expansion of 1/((1+x)*(1-2*x)*(1+x^2)).

1, 1, 2, 4, 9, 17, 34, 68, 137, 273, 546, 1092, 2185, 4369, 8738, 17476, 34953, 69905, 139810, 279620, 559241, 1118481, 2236962, 4473924, 8947849, 17895697, 35791394, 71582788, 143165577, 286331153, 572662306, 1145324612, 2290649225, 4581298449, 9162596898

Offset: 0

Author

Paul Barry, Jan 22 2006

Keywords

• easy
• nonn

Comments

Signed version is A077931. Row sums of A115450.

Links

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

Programs

Mathematica

CoefficientList[Series[1/((1 + x) (1 - 2 x) (1 + x^2)), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
LinearRecurrence[{1,1,1,2},{1,1,2,4},50] (* Harvey P. Dale, Oct 22 2011 *)

PARI

x='x+O('x^50); Vec(1/((1+x)*(1-2*x)*(1+x^2))) \\ G. C. Greubel, Sep 26 2017

Formula

a(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4);

a(n) = (Sum_{k=0..n} (2^(n-k+1)-1)*(-1)^k) - (Sum_{k=0..floor(n/2)} (2^(n-2*k)-1)*(-1)^k).

a(n) = A000975(n+1) - A077854(n-1).

a(n) = A112030(n)/10 + (-1)^n/6 +2^(n+3)/15. - R. J. Mathar, Feb 06 2011