A097072 Expansion of (1 - 2*x + 2*x^2)/((1 - x^2)*(1 - 2*x)).

1, 0, 3, 4, 11, 20, 43, 84, 171, 340, 683, 1364, 2731, 5460, 10923, 21844, 43691, 87380, 174763, 349524, 699051, 1398100, 2796203, 5592404, 11184811, 22369620, 44739243, 89478484, 178956971, 357913940, 715827883, 1431655764, 2863311531, 5726623060, 11453246123

Offset: 0

Author

Paul Barry, Jul 22 2004

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A000035, A000975, A001045, A007583, A080674.

Programs

Magma

[(4*2^n - 3 + 5*(-1)^n)/6: n in [0..50]]; // G. C. Greubel, Oct 10 2017

Maple

a:= n-> ceil(2*(2^n-1)/3)+(-1)^n:
seq(a(n), n=0..32);  # Alois P. Heinz, Jun 15 2023

Mathematica

CoefficientList[Series[(1-2x+2x^2)/((1-x^2)(1-2x)),{x,0,50}],x]  (* Harvey P. Dale, Mar 09 2011 *)
Table[2*2^n/3 - 1/2 + 5 (-1)^n/6, {n, 0, 32}] (* Michael De Vlieger, Feb 22 2017 *)

PARI

for(n=0,50, print1((4*2^n - 3 + 5*(-1)^n)/6, ", ")) \\ G. C. Greubel, Oct 10 2017

Formula

a(n) = (4*2^n - 3 + 5*(-1)^n)/6.

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

a(n) = A001045(n+1) - A000035(n).

a(n) = a(n-1) + 2*a(n-2) + 1, n > 1. - Gary Detlefs, Jun 20 2010

a(2*n) = A007583(n), a(2*n+1) = A080674(n), n >= 0. - Yosu Yurramendi, Feb 21 2017

a(n) = A000975(n) + (-1)^n. - Alois P. Heinz, Jun 15 2023