A099485 -id:A099485 - 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

A099484 A Fibonacci convolution.

Original entry on oeis.org

1, 1, 2, 7, 19, 48, 125, 329, 862, 2255, 5903, 15456, 40465, 105937, 277346, 726103, 1900963, 4976784, 13029389, 34111385, 89304766, 233802911, 612103967, 1602508992, 4195423009, 10983760033, 28755857090, 75283811239, 197095576627

Offset: 0

Paul Barry, Oct 18 2004

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

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-1).

Cf. A000045, A099483, A099485.

LinearRecurrence[{3,-2,3,-1},{1,1,2,7},40] (* Harvey P. Dale, Mar 25 2020 *)

G.f.: (1-x)^2/((1+x^2)*(1-3*x+x^2)), convolution of A176742 and A001906.
a(n)=3a(n-1)-2a(n-2)+3a(n-3);
a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^n*(3^(n-2k)+2*0^(n-2k))/3};
a(n)=sum{k=0..n, (0^k-2sin(pi*k/2))F(2(n-k)+2)}.
(1/3) [Fib(2n+2) + I^n + (-I)^n ]. - Ralf Stephan, Dec 04 2004
3*a(n) = A001906(n+1) +2*A056594(n). - R. J. Mathar, Jun 17 2020

cp's OEIS Frontend

A099483 A Fibonacci convolution.

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