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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.
1
0, 0, 0, 1, 2, 3, 6, 10, 16, 27, 44, 71, 116, 188, 304, 493, 798, 1291, 2090, 3382, 5472, 8855, 14328, 23183, 37512, 60696, 98208, 158905, 257114, 416019, 673134, 1089154, 1762288, 2851443, 4613732, 7465175, 12078908, 19544084, 31622992, 51167077, 82790070
OFFSET
1,5
COMMENTS
See also similar sequence A130578.
Numbers whose Zeckendorf representation is a prefix of 100100100... . - Jeffrey Shallit, Jun 29 2024
FORMULA
G.f.: x^4*(1+x)/((1-x)(1+x+x^2)(1-x-x^2)). - Alois P. Heinz, Dec 13 2011
a(n) = (1/2)*(A000045(n)-A131534(n+1)). - Bruno Berselli, Dec 14 2011
a(n) = F(n) - ceiling(F(n-1)/2) - ceiling(F(n-2)/2). - Chunqing Liu, Aug 21 2023
MAPLE
a:= n-> (Matrix(5, (i, j)-> `if`(i=j-1, 1, `if`(i=5,
[-1, -1, 1, 1, 1][j], 0)))^n. <<-1, 0, 0, 0, 1>>)[1, 1]:
seq(a(n), n=1..50); # Alois P. Heinz, Dec 13 2011
MATHEMATICA
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 *)
PROG
(Magma) [IsOdd(Fibonacci(n)) select (Fibonacci(n)-1)/2 else Fibonacci(n)/2-1: n in [1..41]]; // Bruno Berselli, Dec 14 2011
CROSSREFS
Sequence in context: A023561 A243735 A034419 * A198200 A294444 A066895
KEYWORD
nonn,easy
AUTHOR
Giovanni Teofilatto, Dec 06 2011
STATUS
approved