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%I #21 Mar 23 2024 08:04:20
%S 2,1,1,3,6,8,12,21,35,55,88,144,234,377,609,987,1598,2584,4180,6765,
%T 10947,17711,28656,46368,75026,121393,196417,317811,514230,832040,
%U 1346268,2178309,3524579,5702887,9227464,14930352,24157818,39088169
%N a(n) = Fibonacci(n+1)+cos(n*Pi/2).
%C a(n) is the convolution of L(n) with the sequence (1,0,-1,0,1,0,-1,0,....) A056594. a(2n+1)=F(2n+2), F = Fibonacci numbers.
%C a(n) = Sum((-1)^(i+floor(n/2))L(2i+e),(i=0..floor(n/2))), where L(n) Lucas numbers and e=(1/2)(1-(-1)^n).
%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), a(0)=2, a(1)=1, a(2)=1, a(3)=3.
%F G.f.: (2 - x)/(1 - x - x^3 - x^4).
%F a(4n) = F(4n+1)+1, a(4n+2) = F(4n+3)-1.
%F a(n+1)*a(n+3) = a(n)*a(n+2) + a(n+1)*a(n+2) for all n in Z. - _Michael Somos_, Jan 19 2014
%e G.f. = 2 + x + x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 12*x^6 + 21*x^7 + 35*x^8 + ...
%t CoefficientList[Series[(2 - x)/(1 - x - x^3 - x^4), {x, 0, 40}], x]
%t a[ n_] := Fibonacci[n + 1] + Re[I^n] (* _Michael Somos_, Jan 19 2014 *)
%t LinearRecurrence[{1,0,1,1},{2,1,1,3},50] (* _Harvey P. Dale_, Sep 08 2023 *)
%o (PARI) {a(n) = fibonacci(n+1) + real(I^n)} /* _Michael Somos_, Jan 19 2014 */
%Y Cf. A000045, A056594.
%K easy,nonn
%O 0,1
%A Mario Catalani (mario.catalani(AT)unito.it), Aug 29 2002
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