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A123740 Characteristic sequence for Wythoff AB-numbers A003623.

%I A123740 #39 Mar 29 2023 14:52:02
%S A123740 0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,
%T A123740 0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,
%U A123740 0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,1
%N A123740 Characteristic sequence for Wythoff AB-numbers A003623.
%C A123740 Left shifted sequence is the characteristic function of A035336, and also the second lowest digit of the Zeckendorf expansion of n. - _Franklin T. Adams-Watters_, Jun 30 2009
%C A123740 a(n) = A188009(n+2), n>=1. - _Wolfdieter Lang_, Jun 27 2011
%C A123740 Doubling the 0’s in the infinite Fibonacci word A003849 gives (a(n)). - _Michel Dekking_, Sep 09 2016
%C A123740 This is a morphic sequence, i.e., the letter-to-letter image of the fixed point of a morphism. The fixed point is the unique fixed point A270788 of the three symbol Fibonacci morphism. The letter-to-letter map is 1->0, 2->0, 3->1. - _Michel Dekking_, May 02 2019
%D A123740 See references under A000201.
%H A123740 Michael De Vlieger, <a href="/A123740/b123740.txt">Table of n, a(n) for n = 1..16384</a>
%H A123740 Michel Dekking and Michael Keane, <a href="https://arxiv.org/abs/1608.04487">On the conjugacy class of the Fibonacci dynamical system</a>, arXiv preprint arXiv:1608.04487 [math.DS], 2016.
%H A123740 Michel Dekking and Michael Keane, <a href="https://doi.org/10.1016/j.tcs.2017.01.009"> On the conjugacy class of the Fibonacci dynamical system</a>, Theoretical Computer Science 668 (2017), 59-69.
%H A123740 Jeffrey Shallit and Anatoly Zavyalov, <a href="https://arxiv.org/abs/2303.15203">Transduction of Automatic Sequences and Applications</a>, arXiv:2303.15203 [cs.FL], 2023, see p. 31.
%F A123740 a(n) = 1 if n=A(B(k)) for some k>=1, else 0, with A(k):=A000201(k) and B(k):=A001950(k), k>=1.
%F A123740 a(n) = 1-(1-h(n))-(1-h(n+1)) = h(n)-(1-h(n+1))= h(n)*h(n+1) with h(n):=A005614(n-1), n>=1, the rabbit sequence.
%F A123740 a(n) = A(n+2)-A(n)-3. - _Wolfdieter Lang_, Jun 27 2011
%t A123740 a[_] = 0; s = Table[n + 2 Floor[n*GoldenRatio], {n, 24}]; Map[Set[a[#], 1] &, s]; Array[a, Max[s]] (* _Michael De Vlieger_, Mar 29 2023 *)
%o A123740 (Python)
%o A123740 from math import isqrt
%o A123740 def A123740(n): return (n+2+isqrt(m:=5*(n+2)**2)>>1)-(n+isqrt(m-20*(n+1))>>1)-3 # _Chai Wah Wu_, Aug 29 2022
%Y A123740 Cf. A003623, A000201, A001950, A188009, A270788.
%Y A123740 Cf. A003849, A014417. - _Franklin T. Adams-Watters_, Jun 30 2009
%K A123740 nonn,easy
%O A123740 1,1
%A A123740 _Wolfdieter Lang_, Oct 13 2006