A114199 Row sums of a Pascal-Fibonacci triangle.

1, 2, 4, 8, 17, 38, 87, 200, 458, 1044, 2373, 5388, 12233, 27782, 63112, 143392, 325805, 740266, 1681935, 3821412, 8682310, 19726316, 44818473, 101828344, 231355953, 525645354, 1194276812, 2713420728, 6164945513, 14006877390

Offset: 0

Author

Paul Barry, Nov 16 2005

Keywords

• easy
• nonn

Comments

Binomial transform of double Fibonacci sequence A103609(n+2). Row sums of A114197.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2806
• Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2,1).

Crossrefs

Cf. A000045, A103609, A114197.

Programs

Magma

[n le 4 select 2^(n-1) else 4*Self(n-1) -5*Self(n-2) +2*Self(n-3) +Self(n-4): n in [1..30]]; // G. C. Greubel, Oct 23 2024

Mathematica

LinearRecurrence[{4,-5,2,1},{1,2,4,8},30] (* Harvey P. Dale, Dec 07 2015 *)

SageMath

@CachedFunction # a = A114199
def a(n): return 2^n if n<4 else 4*a(n-1) -5*a(n-2) +2*a(n-3) +a(n-4)
[a(n) for n in range(71)] # G. C. Greubel, Oct 23 2024

Formula

G.f.: (1-x)^2/(1-4*x+5*x^2-2*x^3-x^4).

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n-k, j)*C(k, j)*Fibonacci(j).

a(n) = Sum_{k=0..n} C(n, k)*Fibonacci(floor((k+2)/2)).