OFFSET
0,2
COMMENTS
A090040 is the Q-residue of the triangle A175840, where Q is the triangular array (t(i,j)) given by t(i,j)=1; see A193649 for the definition of Q-residue. - Clark Kimberling, Aug 07 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-12).
FORMULA
G.f.: (1-3*x)/((1-2*x)*(1-6*x)).
E.g.f.: (3*exp(6*x)+exp(2*x))/4 = exp(4*x)*(cosh(2*x)+sinh(2*x)/2).
a(n) = 8*a(n-1) -12*a(n-2), a(0)=1, a(1)=5.
a(n) = (3*6^n+2^n)/4.
a(n)=6*a(n-1)-2^(n-1). - Paul Curtz, Jan 09 2009
Fourth binomial transform of (1, 1, 4, 4, 16, 16, ...). a(n)=sum{k=1..floor(n/2), C(n, 2k)4^(n-k-1)}. - Paul Barry, Nov 22 2003
a(n) = A019590 (mod 4), proof via a(n)=8*a(n-1)-12*a(n-2). - R. J. Mathar, Feb 25 2009
a(n) = Sum_{k, 0<=k<=n} A117317(n,k)*3^k. - Philippe Deléham, Jan 28 2012
MATHEMATICA
LinearRecurrence[{8, -12}, {1, 5}, 30] (* Harvey P. Dale, Nov 23 2014 *)
PROG
(Magma) [(3*6^n+2^n)/4: n in [0..30]]; // Vincenzo Librandi, Jun 10 2011
(PARI) a(n)=(3*6^n+2^n)/4 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 20 2003
STATUS
approved