A167816 Numerator of x(n) = x(n-1) + x(n-2), x(0)=0, x(1)=1/3; denominator=A167817.

0, 1, 1, 2, 1, 5, 8, 13, 7, 34, 55, 89, 48, 233, 377, 610, 329, 1597, 2584, 4181, 2255, 10946, 17711, 28657, 15456, 75025, 121393, 196418, 105937, 514229, 832040, 1346269, 726103, 3524578, 5702887, 9227465, 4976784, 24157817, 39088169, 63245986, 34111385

Offset: 0

Author

Reinhard Zumkeller, Nov 13 2009

Keywords

• frac
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Wikipedia, Fibonacci number
• Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 7, 0, 0, 0, -1).

Crossrefs

Cf. A000045, A167808.

Programs

Magma

[0,1,1] cat [Numerator(Fibonacci(n)/Fibonacci(2*n-4)): n in [3..40]]; // Vincenzo Librandi, Jun 28 2016

Mathematica

Numerator[LinearRecurrence[{1,1},{0,1/3},40]] (* Harvey P. Dale, Dec 07 2014 *)
LinearRecurrence[{0, 0, 0, 7, 0, 0, 0, -1},{0, 1, 1, 2, 1, 5, 8, 13},39] (* Ray Chandler, Aug 03 2015 *)

Formula

a(n) = (a(n-1)*A093148(n+2) + a(n-2)*A093148(n+1))/A093148(n-1) for n>1.

a(4*n) = A004187(n) = (a(4*n-1) + a(4*n-2))/3;

a(4*n+1) = A033889(n) = 3*a(4*n-1) + a(4*n-2);

a(4*n+2) = A033890(n) = a(4*n-1) + 3*a(4*n-2);

a(4*n+3) = A033891(n) = a(4*n-1) + a(4*n-2).

Numerator of Fibonacci(n) / Fibonacci(2n-4) for n>=3. - Gary Detlefs, Dec 20 2010

Extensions

Definition corrected by D. S. McNeil, May 09 2010