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A099485
A Fibonacci convolution.
3
1, 2, 5, 14, 37, 96, 251, 658, 1723, 4510, 11807, 30912, 80929, 211874, 554693, 1452206, 3801925, 9953568, 26058779, 68222770, 178609531, 467605822, 1224207935, 3205017984, 8390846017, 21967520066, 57511714181, 150567622478
OFFSET
0,2
COMMENTS
A Chebyshev transform of A025192 with g.f. (1-x)/(1-3*x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))*G(x/(1+x^2)).
LINKS
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics. 2017, Vol. 41 Issue 6, pp. 849-853.
FORMULA
G.f.: (1-x+x^2)/((1+x^2)*(1-3*x+x^2)).
a(n) = 3*a(n-1)-2*a(n-2)+3*a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^n*(2*3^(n-2*k)+0^(n-2*k))/3.
a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*Fibonacci(2*(n-k)+2).
a(n) = (1/6) * (4*Fibonacci(2*n+2) + I^n + (-I)^n). - Ralf Stephan, Dec 04 2004
Also a transformation of the Jacobsthal numbers A001045(n+1) under the mapping G(x)-> (1/(1-x+x^2))*G(x/(1-x+x^2)). - Paul Barry, Dec 11 2004
G.f.: g(f(x))/x, where g is g.f. of A001045 and f is g.f. of A128834. - Oboifeng Dira, Jun 21 2020
MATHEMATICA
LinearRecurrence[{3, -2, 3, -1}, {1, 2, 5, 14}, 30] (* Harvey P. Dale, Jul 06 2017 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 18 2004
STATUS
approved