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A099484
A Fibonacci convolution.
3
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
0,3
COMMENTS
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)).
FORMULA
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
MATHEMATICA
LinearRecurrence[{3, -2, 3, -1}, {1, 1, 2, 7}, 40] (* Harvey P. Dale, Mar 25 2020 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 18 2004
STATUS
approved