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

0,3

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).

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]

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 *)

Prog

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

Crossrefs

Sequence in context: A272227 A272587 A125069 * A184189 A152098 A041673

Adjacent sequences: A080923 A080924 A080925 * A080927 A080928 A080929

Keyword

nonn,easy

Author

Paul Barry, Feb 26 2003

Status

approved