%I #26 Nov 05 2024 05:40:17
%S 2,2,1,2,6,9,12,20,35,56,88,143,234,378,609,986,1598,2585,4180,6764,
%T 10947,17712,28656,46367,75026,121394,196417,317810,514230,832041,
%U 1346268,2178308,3524579,5702888,9227464,14930351,24157818,39088170,63245985,102334154
%N a(n) = F(floor( (n+3)/2 )) * L(floor( (n+2)/2 )) where F=Fibonacci and L=Lucas numbers.
%H G. C. Greubel, <a href="/A236144/b236144.txt">Table of n, a(n) for n = 0..2500</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,1).
%F G.f.: (2 - x^2 - x^3) / (1 - x - x^3 - x^4) = (1 - x) * (2 + 2*x + x^2) / ((1 + x^2) * (1 - x - x^2)).
%F a(n) = a(n-1) + a(n-3) + a(n-4) for all n in Z.
%F 0 = a(n)*a(n+2) + a(n+1)*(+a(n+2) -a(n+3)) for all n in Z.
%F a(n) = A115008(n+2) - A115008(n+1).
%F a(n) = A115339(n) * A115339(n-1).
%F a(2*n - 1) = F(n+1) * L(n-1) = A128535(n+1). a(2*n) = F(n+1) * L(n) = A128534(n+1).
%F a(n) = A000045(n+1)+A057077(n). - _R. J. Mathar_, Sep 24 2021
%e G.f. = 2 + 2*x + x^2 + 2*x^3 + 6*x^4 + 9*x^5 + 12*x^6 + 20*x^7 + 35*x^8 + ...
%t a[ n_] := Fibonacci[ Quotient[ n + 3, 2]] LucasL[ Quotient[ n, 2]];
%t CoefficientList[Series[(2-x^2-x^3)/(1-x-x^3-x^4), {x, 0, 60}], x] (* _G. C. Greubel_, Aug 07 2018 *)
%o (PARI) {a(n) = fibonacci( (n+3)\2 ) * (fibonacci( n\2+1 ) + fibonacci( n\2-1 ))};
%o (PARI) x='x+O('x^60); Vec((2-x^2-x^3)/(1-x-x^3-x^4)) \\ _G. C. Greubel_, Aug 07 2018
%o (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(2-x^2-x^3)/(1-x-x^3-x^4)); // _G. C. Greubel_, Aug 07 2018
%Y Cf. A000032, A000045, A115008, A115339, A128534, A128535.
%K nonn,easy,changed
%O 0,1
%A _Michael Somos_, Jan 19 2014