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A201864 a(n) = ((F(n-1)+F(n-2))-1)/2 if F(n) is odd, otherwise a(n) = ((F(n-1)+F(n-2))-2)/2, where F(n) = A000045(n) is the n-th Fibonacci number.

%I A201864 #52 Jun 30 2024 09:10:24
%S A201864 0,0,0,1,2,3,6,10,16,27,44,71,116,188,304,493,798,1291,2090,3382,5472,
%T A201864 8855,14328,23183,37512,60696,98208,158905,257114,416019,673134,
%U A201864 1089154,1762288,2851443,4613732,7465175,12078908,19544084,31622992,51167077,82790070
%N A201864 a(n) = ((F(n-1)+F(n-2))-1)/2 if F(n) is odd, otherwise a(n) = ((F(n-1)+F(n-2))-2)/2, where F(n) = A000045(n) is the n-th Fibonacci number.
%C A201864 See also similar sequence A130578.
%C A201864 Numbers whose Zeckendorf representation is a prefix of 100100100... . - _Jeffrey Shallit_, Jun 29 2024
%H A201864 Vincenzo Librandi, <a href="/A201864/b201864.txt">Table of n, a(n) for n = 1..1000</a>
%H A201864 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,-1).
%F A201864 G.f.: x^4*(1+x)/((1-x)(1+x+x^2)(1-x-x^2)). - _Alois P. Heinz_, Dec 13 2011
%F A201864 a(n) = (1/2)*(A000045(n)-A131534(n+1)). - _Bruno Berselli_, Dec 14 2011
%F A201864 a(n) = F(n) - ceiling(F(n-1)/2) - ceiling(F(n-2)/2). - _Chunqing Liu_, Aug 21 2023
%p A201864 a:= n-> (Matrix(5, (i, j)-> `if`(i=j-1, 1, `if`(i=5,
%p A201864         [-1, -1, 1, 1, 1][j], 0)))^n. <<-1, 0, 0, 0, 1>>)[1, 1]:
%p A201864 seq(a(n), n=1..50);  # _Alois P. Heinz_, Dec 13 2011
%t A201864 CoefficientList[Series[x^3*(1+x)/((1-x)(1+x+x^2)(1-x-x^2)),{x,0,30}],x] (* _Vincenzo Librandi_, Mar 20 2012 *)
%o A201864 (Magma) [IsOdd(Fibonacci(n)) select (Fibonacci(n)-1)/2 else Fibonacci(n)/2-1: n in [1..41]];  // _Bruno Berselli_, Dec 14 2011
%Y A201864 Cf. A000045, A131534.
%K A201864 nonn,easy
%O A201864 1,5
%A A201864 _Giovanni Teofilatto_, Dec 06 2011