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<a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2,1).
G.f.: (1-x)^2/(1-4x4*x+5x5*x^2-2x2*x^3-x^4);.
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n-k, j)*C(k, j)*FFibonacci(j);.
a(n) = Sum_{k=0..n} C(n, k)*FFibonacci(floor((k+2)/2)).
(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)
[a(n) for n in range(71)] # G. C. Greubel, Oct 23 2024
Cf. A000045.
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Sergio Falcón, <a href="httphttps://doiwww.rgnpublications.orgcom/journals/10index.26713php/cma.v10i3./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.
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G.f.: (1-x)^2/(1-4x+5x^2-2x^3-x^4); a(n)=sum{k=0..n, sum{j=0..n-k, C(n-k, j)C(k, j)F(j)}}; a(n)=sum{k=0..n, C(n, k)F(floor((k+2)/2))}.
G.f.: (1-x)^2/(1-4x+5x^2-2x^3-x^4);
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n-k, j)*C(k, j)*F(j);
a(n) = Sum_{k=0..n} C(n, k)*F(floor((k+2)/2)).
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