A010056 Characteristic function of Fibonacci numbers: a(n) = 1 if n is a Fibonacci number, otherwise 0.

1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Understood as a binary number, Sum_{k>=0} a(k)/2^k, the resulting decimal expansion is 1.910278797207865891... = Fibonacci_binary+0.5 (see A084119) or Fibonacci_binary_constant-0.5 (see A124091), respectively. - Hieronymus Fischer, May 14 2007

a(n)=1 if and only if there is an integer m such that x=n is a root of p(x)=25*x^4-10*m^2*x^2+m^4-16. Also a(n)=1 iff floor(s)<>floor(c) or ceiling(s)<>ceiling(c) where s=arcsinh(sqrt(5)*n/2)/log(phi), c=arccosh(sqrt(5)*n/2)/log(phi) and phi=(1+sqrt(5))/2. - Hieronymus Fischer, May 17 2007

a(A000045(n)) = 1; a(A001690(n)) = 0. - Reinhard Zumkeller, Oct 10 2013

Image, under the map sending a,b,c -> 1, d,e,f -> 0, of the fixed point, starting with a, of the morphism sending a -> ab, b -> c, c -> cd, d -> d, e -> ef, f -> e. - Jeffrey Shallit, May 14 2016

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.
• D. Bailey et al., On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux (2004), Volume: 16, Issue: 3, page 487-518.
• Wikipedia, Fibonacci number
• Index entries for characteristic functions

Crossrefs

Cf. A000045, A001690, A072649, A104162, A108852, A124091, A130233, A130234.

Decimal expansion of Fibonacci binary is in A084119.

Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

Cf. A079586 (Dirich. g.f. at s=1).

Programs

Haskell

import Data.List (genericIndex)
a010056 = genericIndex a010056_list
a010056_list = 1 : 1 : ch [2..] (drop 3 a000045_list) where
   ch (x:xs) fs'@(f:fs) = if x == f then 1 : ch xs fs else 0 : ch xs fs'
-- Reinhard Zumkeller, Oct 10 2013

Maple

a:= n-> (t-> `if`(issqr(t+4) or issqr(t-4), 1, 0))(5*n^2):
seq(a(n), n=0..144);  # Alois P. Heinz, Dec 06 2020

Mathematica

Join[{1},With[{fibs=Fibonacci[Range[15]]},If[MemberQ[fibs,#],1,0]& /@Range[100]]]  (* Harvey P. Dale, May 02 2011 *)

PARI

a(n)=my(k=n^2);k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)) \\ Charles R Greathouse IV, Jul 30 2012

Python

from sympy.ntheory.primetest import is_square
def A010056(n): return int(is_square(m:=5*n**2-4) or is_square(m+8)) # Chai Wah Wu, Mar 30 2023

Formula

G.f.: (Sum_{k>=0} x^A000045(k)) - x. - Hieronymus Fischer, May 17 2007