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.

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

Author

Giovanni Teofilatto, Dec 06 2011

Keywords

• nonn
• easy

Comments

See also similar sequence A130578.

Numbers whose Zeckendorf representation is a prefix of 100100100... . - Jeffrey Shallit, Jun 29 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).

Crossrefs

Cf. A000045, A131534.

Programs

Magma

[IsOdd(Fibonacci(n)) select (Fibonacci(n)-1)/2 else Fibonacci(n)/2-1: n in [1..41]];  // Bruno Berselli, Dec 14 2011

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 *)

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