A080926 Partial sums of A080925.

0, 1, 6, 19, 60, 181, 546, 1639, 4920, 14761, 44286, 132859, 398580, 1195741, 3587226, 10761679, 32285040, 96855121, 290565366, 871696099, 2615088300, 7845264901, 23535794706, 70607384119, 211822152360, 635466457081

Offset: 0

Author

Paul Barry, Feb 26 2003

Keywords

• nonn
• easy

Comments

This is the sequence A(0,1;2,3;4) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [Wolfdieter Lang, Oct 18 2010]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [_Wolfdieter Lang_, Oct 18 2010]
• Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Programs

Magma

[(3*3^n+(-1)^n)/4-1: n in [0..30]]; // Vincenzo Librandi, Aug 06 2013

Maple

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+4 od: seq(a[n], n=0..33); # Zerinvary Lajos, Dec 14 2008

Mathematica

CoefficientList[Series[x (1 + 3 x) / ((1 + x) (1 - x) (1 - 3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{3,1,-3},{0,1,6},30] (* Harvey P. Dale, Oct 02 2018 *)

Formula

a(n) = Sum{i=0..n, Sum{k=1..i, Binomial(i, 2k-2)2^(2k-2)}}

G.f.: x*(1+3*x)/((1+x)*(1-x)*(1-3x)).

E.g.f.: (3*exp(3x)+exp(-x))/4-exp(x).

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

a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n>2, a(0)=0, a(1)=1, a(2)=6. Observation by G. Detlefs. See my comment and link. [Wolfdieter Lang, Oct 18 2010]