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A099485 A Fibonacci convolution.

%I A099485 #45 Jun 27 2020 14:20:39
%S A099485 1,2,5,14,37,96,251,658,1723,4510,11807,30912,80929,211874,554693,
%T A099485 1452206,3801925,9953568,26058779,68222770,178609531,467605822,
%U A099485 1224207935,3205017984,8390846017,21967520066,57511714181,150567622478
%N A099485 A Fibonacci convolution.
%C A099485 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)).
%H A099485 Michael De Vlieger, <a href="/A099485/b099485.txt">Table of n, a(n) for n = 0..2392</a>
%H A099485 Oboifeng Dira, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_201706&amp;filename=07_41(6).pdf">A Note on Composition and Recursion</a>, Southeast Asian Bulletin of Mathematics. 2017, Vol. 41 Issue 6, pp. 849-853.
%H A099485 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,3,-1).
%F A099485 G.f.: (1-x+x^2)/((1+x^2)*(1-3*x+x^2)).
%F A099485 a(n) = 3*a(n-1)-2*a(n-2)+3*a(n-3).
%F A099485 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^n*(2*3^(n-2*k)+0^(n-2*k))/3.
%F A099485 a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*Fibonacci(2*(n-k)+2).
%F A099485 a(n) = (1/6) * (4*Fibonacci(2*n+2) + I^n + (-I)^n). - _Ralf Stephan_, Dec 04 2004
%F A099485 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
%F A099485 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
%t A099485 LinearRecurrence[{3,-2,3,-1},{1,2,5,14},30] (* _Harvey P. Dale_, Jul 06 2017 *)
%Y A099485 Cf. A000045, A099483, A099484, A001045, A128834.
%K A099485 easy,nonn
%O A099485 0,2
%A A099485 _Paul Barry_, Oct 18 2004