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a(n) = Sum_{i = 0..floor(n/2)} (-1)^(i + floor(n/2)) F(2*i + e), where F = A000045 (Fibonacci numbers) and e = (1-(-1)^n)/2.
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%I #26 Jan 15 2024 16:26:34

%S 0,1,1,1,2,4,6,9,15,25,40,64,104,169,273,441,714,1156,1870,3025,4895,

%T 7921,12816,20736,33552,54289,87841,142129,229970,372100,602070,

%U 974169,1576239,2550409,4126648,6677056,10803704,17480761,28284465,45765225,74049690

%N a(n) = Sum_{i = 0..floor(n/2)} (-1)^(i + floor(n/2)) F(2*i + e), where F = A000045 (Fibonacci numbers) and e = (1-(-1)^n)/2.

%C Essentially the same as A006498 (g.f. 1/(1-x-x^3-x^4)).

%C a(n) is the convolution of F(n) with the sequence (1,0,-1,0,1,0,-1,0,...), A056594.

%H Reinhard Zumkeller, <a href="/A074677/b074677.txt">Table of n, a(n) for n = 0..1000</a>

%H Victoria Zhuravleva, <a href="http://dx.doi.org/10.5802/jtnb.846">Diophantine approximations with Fibonacci numbers</a>, Journal de théorie des nombres de Bordeaux, 25 no. 2 (2013), p. 499-520. See Lemma 5.1.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,1).

%F a(n) = a(n-1) + a(n-3) + a(n-4) for n>3, a(0)=0, a(1)=1, a(2)=1, a(3)=1.

%F G.f.: x/(1 - x - x^3 - x^4).

%F a(n) = Fibonacci(ceiling(n/2))*Fibonacci(floor(n/2+1)). - _Alois P. Heinz_, Jan 15 2024

%t CoefficientList[Series[x/(1 - x - x^3 - x^4), {x, 0, 40}], x]

%o (Haskell)

%o a074677 n = a074677_list !! (n-1)

%o a074677_list = 0 : 1 : 1 : 1 : zipWith (+) a074677_list

%o (zipWith (+) (tail a074677_list) (drop 3 a074677_list))

%o -- _Reinhard Zumkeller_, Dec 28 2011

%o (Sage) [sum((-1)^(i+floor(n/2))*fibonacci(2*i+(1-(-1)^n)/2) for i in (0..floor(n/2))) for n in [0..50]]; # _Bruno Berselli_, Mar 15 2016

%o (Magma) [&+[(-1)^(i+Floor(n/2))*Fibonacci(2*i+(1-(-1)^n) div 2): i in [0..Floor(n/2)]]: n in [0..50]]; // _Bruno Berselli_, Mar 15 2016

%o (PARI) concat(0, Vec(x/((1+x^2)*(1-x-x^2)) + O(x^50))) \\ _Colin Barker_, Mar 15 2016

%Y Cf. A000045, A056594.

%K nonn,easy

%O 0,5

%A Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2002