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A010056 a(n) = 1 if n is a Fibonacci number, otherwise 0. 26
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

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.

Index entries for characteristic functions

FORMULA

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

MATHEMATICA

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

PROG

(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

(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

CROSSREFS

Cf. A000045, A001690, A084119, A104162, A108852, A124091, A130233, A130234.

Decimal expansion of Fibonacci binary is in A084119.

Sequence in context: A121802 A156241 A156254 * A155898 A181650 A115952

Adjacent sequences:  A010053 A010054 A010055 * A010057 A010058 A010059

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 18 14:01 EDT 2017. Contains 290720 sequences.