A114199 - OEIS

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

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OFFSET

0,2

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).

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)).

MATHEMATICA

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

PROG

(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

(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)