A097164 Expansion of (1+3x)/((1-x)(1-4x^2)).

1, 4, 8, 20, 36, 84, 148, 340, 596, 1364, 2388, 5460, 9556, 21844, 38228, 87380, 152916, 349524, 611668, 1398100, 2446676, 5592404, 9786708, 22369620, 39146836, 89478484, 156587348, 357913940, 626349396, 1431655764, 2505397588

Offset: 0

Author

Paul Barry, Jul 30 2004

Keywords

• easy
• nonn

Comments

Partial sums of A084221. a(n) = A097163(n+1)/4. Third binomial transform is A097165.

a(n+1) = 4*A097163(n). - Zerinvary Lajos, Mar 17 2008

See A133628 for an essentially identical sequence. - R. J. Mathar, Jun 08 2008

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], n=1..31); # Zerinvary Lajos, Mar 17 2008

Mathematica

CoefficientList[Series[(1+3x)/((1-x)(1-4x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{1,4,-4},{1,4,8},50] (* Harvey P. Dale, Jul 11 2023 *)

Formula

a(n) = 5*2^n/2 - (-2)^n/6 - 4/3;

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

G.f. ( 1+3*x ) / ( (x-1)*(2*x+1)*(2*x-1) ). - R. J. Mathar, Jul 06 2011