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 A010056 Characteristic function of Fibonacci numbers: a(n) = 1 if n is a Fibonacci number, otherwise 0. 53
 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 Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807, Nov 29 2017 D. Bailey et al., On the binary expansions of algebraic numbers, Journal de ThÃ©orie des Nombres de Bordeaux (2004), Volume: 16, Issue: 3, page 487-518. 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. 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. Sequence in context: A121802 A156241 A156254 * A155898 A181650 A115952 Adjacent sequences:  A010053 A010054 A010055 * A010057 A010058 A010059 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified August 18 00:28 EDT 2018. Contains 313817 sequences. (Running on oeis4.)