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Row sums of a Pascal-Fibonacci triangle.
2

%I #19 Oct 25 2024 07:14:23

%S 1,2,4,8,17,38,87,200,458,1044,2373,5388,12233,27782,63112,143392,

%T 325805,740266,1681935,3821412,8682310,19726316,44818473,101828344,

%U 231355953,525645354,1194276812,2713420728,6164945513,14006877390

%N Row sums of a Pascal-Fibonacci triangle.

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

%H Michael De Vlieger, <a href="/A114199/b114199.txt">Table of n, a(n) for n = 0..2806</a>

%H Sergio Falcón, <a href="https://www.rgnpublications.com/journals/index.php/cma/article/viewFile/1221/950">Binomial Transform of the Generalized k-Fibonacci Numbers</a>, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2,1).

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

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

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

%t LinearRecurrence[{4,-5,2,1},{1,2,4,8},30] (* _Harvey P. Dale_, Dec 07 2015 *)

%o (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

%o (SageMath)

%o @CachedFunction # a = A114199

%o 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)

%o [a(n) for n in range(71)] # _G. C. Greubel_, Oct 23 2024

%Y Cf. A000045, A103609, A114197.

%K easy,nonn,changed

%O 0,2

%A _Paul Barry_, Nov 16 2005