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 (seeA084119) or Fibonacci_binary_constant-0.5 (seeA124091), 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
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
FORMULA
G.f.: (Sum_{k>=0} x^A000045(k)) - x. -Hieronymus Fischer,May 17 2007
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 *)
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
(Python)
from sympy.ntheory.primetest import is_square
defA010056(n): return int(is_square(m:=5*n**2-4) or is_square(m+8)) #Chai Wah Wu,Mar 30 2023
CROSSREFS
Decimal expansion of Fibonacci binary is inA084119.
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1:A003849,2:A010060,3:A010056,4:A020985andA020987,5:A191818,6:A316340andA273129,18:A316341,19:A030302,20:A063438,21:A316342,22:A316343,23:A003849minus its first term, 24:A316344,25:A316345andA316824,26:A020985andA020987,27: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).
KEYWORD
nonn,easy
AUTHOR
STATUS
approved