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
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Feb 26 2003
STATUS
approved