A104487 a(n+3) = 6a(n+2) - 10a(n+1) + 3a(n); a(0) = 1, a(1) = 4, a(2) = 14.

1, 4, 14, 47, 154, 496, 1577, 4964, 15502, 48103, 148490, 456416, 1397905, 4268740, 13002638, 39522143, 119912698, 363262672, 1099015481, 3321204260, 10026858766, 30246156439, 91171963754, 274650794432, 826923598369

Offset: 0

Author

Creighton Dement, Apr 19 2005

Keywords

• nonn
• easy

Comments

Binomial transform of A104004.

If another a(0)=0 is added in front, also the binomial transform of A027934.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-10,3).

Programs

Magma

[3^(n+1) - Fibonacci(2*n+3): n in [0..30]]; // Vincenzo Librandi, Apr 21 2011

Mathematica

LinearRecurrence[{6,-10,3},{1,4,14},30] (* Harvey P. Dale, May 07 2017 *)

Formula

G.f.: (2*x-1)/((3*x-1)*(x^2-3*x+1)). Define c = (3+sqrt(5))/2 and d = (3-sqrt(5))/2. Then a(n) = 3^(n+1) - ((2*sqrt(5)/5)+1)*c^n + ((2*sqrt(5)/5)-1)*d^n = 3^(n+1) - Fibonacci(2n+3). - Ralf Stephan, May 20 2007

Extensions

Comment concerning the binomial transforms corrected by R. J. Mathar, Oct 26 2009