A099484 -id:A099484 - cp's OEIS frontend

A099483 A Fibonacci convolution.

0, 1, 3, 7, 18, 48, 126, 329, 861, 2255, 5904, 15456, 40464, 105937, 277347, 726103, 1900962, 4976784, 13029390, 34111385, 89304765, 233802911, 612103968, 1602508992, 4195423008, 10983760033, 28755857091, 75283811239, 197095576626

Offset: 0

Paul Barry, Oct 18 2004

A Chebyshev transform of the sequence 0,1,3,9,27 with g.f. x/(1-3x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-1).

Cf. A000045, A099484, A099485.

LinearRecurrence[{3,-2,3,-1},{0,1,3,7},30] (* Harvey P. Dale, May 23 2016 *)

G.f.: x/((1+x^2)(1-3x+x^2)); a(n)=3a(n-1)-2a(n-2)+3a(n-3); a(n)=sum{k=0..n, cos(pi*k/2)F(2(n-k))}. a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^n*(3^(n-2k)-0^(n-2k))/3}.

(1/6) [2Fib(2n+2) - I^n - (-I)^n ]. - Ralf Stephan, Dec 04 2004

A099485 A Fibonacci convolution.

Original entry on oeis.org

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

Paul Barry, Oct 18 2004

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)).

Michael De Vlieger, Table of n, a(n) for n = 0..2392
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics. 2017, Vol. 41 Issue 6, pp. 849-853.
Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-1).

Cf. A000045, A099483, A099484, A001045, A128834.

LinearRecurrence[{3,-2,3,-1},{1,2,5,14},30] (* Harvey P. Dale, Jul 06 2017 *)

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

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A099483 A Fibonacci convolution.

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