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A099485 A Fibonacci convolution. 3

%I #45 Jun 27 2020 14:20:39

%S 1,2,5,14,37,96,251,658,1723,4510,11807,30912,80929,211874,554693,

%T 1452206,3801925,9953568,26058779,68222770,178609531,467605822,

%U 1224207935,3205017984,8390846017,21967520066,57511714181,150567622478

%N A Fibonacci convolution.

%C 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 Michael De Vlieger, <a href="/A099485/b099485.txt">Table of n, a(n) for n = 0..2392</a>

%H 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,3,-1).

%F G.f.: (1-x+x^2)/((1+x^2)*(1-3*x+x^2)).

%F a(n) = 3*a(n-1)-2*a(n-2)+3*a(n-3).

%F 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 a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*Fibonacci(2*(n-k)+2).

%F a(n) = (1/6) * (4*Fibonacci(2*n+2) + I^n + (-I)^n). - _Ralf Stephan_, Dec 04 2004

%F 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 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 LinearRecurrence[{3,-2,3,-1},{1,2,5,14},30] (* _Harvey P. Dale_, Jul 06 2017 *)

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

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 18 2004

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)