A103609 Fibonacci numbers repeated (cf. A000045).

0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 8, 13, 13, 21, 21, 34, 34, 55, 55, 89, 89, 144, 144, 233, 233, 377, 377, 610, 610, 987, 987, 1597, 1597, 2584, 2584, 4181, 4181, 6765, 6765, 10946, 10946, 17711, 17711, 28657, 28657, 46368, 46368, 75025, 75025, 121393

Offset: 0

Author

Roger L. Bagula, Mar 24 2005

Keywords

• nonn
• easy

Comments

The usual policy in the OEIS is not to include such "doubled" sequences. This is an exception. - N. J. A. Sloane

The Gi2 sums, see A180662, of triangle A065941 equal the terms of this sequence without the two leading zeros. - Johannes W. Meijer, Aug 16 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
• N. J. A. Sloane, 2178 And All That [Local copy]
• I. Wloch, U. Bednarz, D. Bród, A Wloch and M. Wolowiec-Musial, On a new type of distance Fibonacci numbers, Discrete Applied Math., Volume 161, Issues 16-17, November 2013, Pages 2695-2701.
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Crossrefs

Partial sums: A094707.

Cf. A000045, A065941, A180662, A214927.

Programs

Magma

[Fibonacci(Floor(n/2)): n in [0..60]]; // G. C. Greubel, Oct 22 2024

Maple

A103609 := proc(n): combinat[fibonacci](floor(n/2)) ; end proc: seq(A103609(n), n=0..52); # Johannes W. Meijer, Aug 16 2011

Mathematica

a[0] = 0; a[1] = 0; a[2] = 1; a[3] = 1; a[n_Integer?Positive] := a[n] = a[n - 2] + a[n - 4]; aa = Table[a[n], {n, 0, 200}]
Join[{0, 0}, LinearRecurrence[{0, 1, 0, 1}, {1, 1, 1, 1}, 60]] (* Vincenzo Librandi, Jan 19 2016 *)
With[{fibs=Fibonacci[Range[0,30]]},Riffle[fibs,fibs]] (* Harvey P. Dale, Jul 11 2025 *)

PARI

a(n)=fibonacci(n\2) \\ Charles R Greathouse IV, Oct 07 2015

PARI

my(x='x+O('x^50)); Vec(x^2*(1+x)/(1-x^2-x^4)) \\ G. C. Greubel, May 01 2017

SageMath

[fibonacci(n//2) for n in range(61)] # G. C. Greubel, Oct 22 2024

Formula

a(n) = a(n-2) + a(n-4).

G.f.: x^2*(1+x)/(1-x^2-x^4). - R. J. Mathar, Sep 27 2008

a(n) = A000045(floor(n/2)). - Johannes W. Meijer, Aug 16 2011

Extensions

Edited by N. J. A. Sloane, Dec 01 2006

Incorrect formula deleted by Johannes W. Meijer, Aug 16 2011