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

0,4

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).

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(2*n-4) for n>=3. - Gary Detlefs, Dec 20 2010

From Elmo R. Oliveira, May 02 2026: (Start)

a(n) = 7*a(n-4) - a(n-8).

G.f.: x*(1+x+2*x^2+x^3-2*x^4+x^5-x^6) / ((-1-x+x^2) * (-1+x+x^2) * (1+3*x^2+x^4)). (End)

Mathematica

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

Prog

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

Crossrefs

Cf. A000045, A167808, A167817.

Cf. A004187, A033889, A033890, A033891, A093148.

Sequence in context: A377363 A193180 A201743 * A316292 A222542 A318052

Adjacent sequences: A167813 A167814 A167815 * A167817 A167818 A167819

Keyword

frac,nonn,easy

Author

Reinhard Zumkeller, Nov 13 2009

Extensions

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

Status

approved