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%I #42 Jun 24 2022 04:44:58
%S 0,1,1,3,4,8,12,21,33,55,88,144,232,377,609,987,1596,2584,4180,6765,
%T 10945,17711,28656,46368,75024,121393,196417,317811,514228,832040,
%U 1346268,2178309,3524577,5702887,9227464,14930352,24157816,39088169
%N a(n) = Fibonacci(n+1) - (1 + (-1)^n)/2.
%C a(n) is the convolution of F(n) with the sequence (1,0,1,0,1,0,...).
%C Transform of F(n) under the Riordan array (1/(1-x^2), x). - _Paul Barry_, Apr 16 2005
%H G. C. Greubel, <a href="/A074331/b074331.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-1).
%F a(n) = Sum_{i=0..floor(n/2)} Fibonacci(2*i + e), where e = 2*(n/2 - floor(n/2)).
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=1, a(3)=3.
%F G.f.: x / ( (1-x)*(1+x)*(1-x-x^2) ).
%F a(2*n+1) = Fibonacci(2*n+2).
%F a(2*n) = Fibonacci(2*n+1) - 1.
%F a(n-1) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1). - _Paul Barry_, Jul 07 2004
%F a(n) = Sum_{k=0..floor((n-1)/2)} Fibonacci(n-2*k). - _Paul Barry_, Apr 16 2005
%F a(n) = Sum_{k=0..n} Fibonacci(k)*(1-(-1)^floor((n+k-1)/2)). - _Paul Barry_, Apr 16 2005
%F a(n) = Fibonacci(n) + a(n-2) for n > 1. - _Zerinvary Lajos_, Mar 17 2008
%F a(n) = floor(g^(n+1)/sqrt(5)), where g = (sqrt(5) + 1)/2. - _Federico Provvedi_, Mar 27 2013
%F E.g.f.: exp(x/2)*(cosh(sqrt(5)*x/2) + (1/sqrt(5))*sinh(sqrt(5)*x/2)) - cosh(x). - _G. C. Greubel_, Jun 23 2022
%p with(combinat):seq(fibonacci(n+1)-(1+(-1)^n)/2, n=0..40); # _Zerinvary Lajos_, Mar 17 2008
%t CoefficientList[Series[x/(1-x-2*x^2+x^3+x^4), {x, 0, 40}], x]
%t Table[Floor[GoldenRatio^(k+1)/Sqrt[5]], {k, 0, 40}] (* _Federico Provvedi_, Mar 26 2013 *)
%o (PARI) a(n)=if(n<0,0,fibonacci(n+1)-(n+1)%2)
%o (Magma) [Fibonacci(n+1) - (1+(-1)^n)/2: n in [0..40]]; // _G. C. Greubel_, Jun 23 2022
%o (SageMath) [fibonacci(n+1) -((n+1)%2) for n in (0..40)] # _G. C. Greubel_, Jun 23 2022
%Y Essentially the same as A052952.
%Y Cf. A000045.
%K nonn,easy
%O 0,4
%A Mario Catalani (mario.catalani(AT)unito.it), Aug 21 2002
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