A097163 Expansion of (1+x-x^2)/((1-x)*(1-4*x^2)).

1, 2, 5, 9, 21, 37, 85, 149, 341, 597, 1365, 2389, 5461, 9557, 21845, 38229, 87381, 152917, 349525, 611669, 1398101, 2446677, 5592405, 9786709, 22369621, 39146837, 89478485, 156587349, 357913941, 626349397, 1431655765, 2505397589

Offset: 0

Author

Paul Barry, Jul 30 2004

Keywords

• nonn
• easy

Comments

Interleave (4*4^n-1)/2 (see A002450) and (7*4^n-1)/3 (A206374).

Links

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

Programs

Maple

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=4*a[n-2]+4 od: seq(a[n]/4, n=2..33); # Zerinvary Lajos, Mar 17 2008

Mathematica

CoefficientList[Series[(1+x-x^2)/((1-x)(1-4x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{1,4,-4},{1,2,5},41] (* or *) f[n_]:=(15*2^n-(-2)^n - 8)/24; Array[f, 40] (* Harvey P. Dale, Jun 17 2011 *)

Formula

G.f.: (1+x-x^2)/((1-x)*(1-4*x^2)).

a(n) = 5*2^n/4+(-2)^n/12-1/3.

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

a(2*n) = A002450(n+1).

a(n) = A097164(n+1)/4.

a(n) = (15*2^n-(-2)^n-8)/24. - Harvey P. Dale, Jun 17 2011