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A095076 Parity of 1-fibits in Zeckendorf expansion A014417(n). 10

%I

%S 0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0,1,1,0,1,1,1,0,

%T 1,0,0,0,1,0,1,1,0,1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0,1,1,0,1,1,1,0,

%U 0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0,0,1,0,1,1,0,1,1,1,0

%N Parity of 1-fibits in Zeckendorf expansion A014417(n).

%C Let u = A000201 = (lower Wythoff sequence) and v = A001950 = (upper Wythoff sequence. Conjecture: This sequence is the sequence p given by p(1) = 0 and p(u(k)) = p(k); p(v(k)) = 1-p(k). [_Clark Kimberling_, Apr 15 2011]

%C [base 2] 0.111010010001100... = 0.9105334708635617... [_Joerg Arndt_, May 13 2011]

%C From _Michel Dekking_, Nov 28 2019: (Start)

%C This sequence is a morphic sequence.

%C Let the morphism sigma be given by

%C sigma(1) = 12, sigma(2) = 4, sigma(3) = 1, sigma(4) = 43,

%C and let the letter-to-letter map lambda be given by

%C lambda(1) = 0, lambda(2) = 1, lambda(3) = 0, lambda(4) = 1.

%C Then a(n) = lambda(x(n)), where x(0)x(1)... = 1244343... is the fixed point of sigma starting with 1.

%C See footnote number 4 in the paper by Emmanuel Ferrand. (End)

%C From _Michel Dekking_, Nov 29 2019: (Start)

%C Proof of Kimberling's conjecture, by virtue of the four symbol morphism sigma in the previous comment.

%C We first show that the fixed point x = 1244343143112... of sigma has the remarkable property that the letters 1 and 4 exclusively occur at positions u(k), k=1,2,..., and the letters 2 and 3 exclusively occur at positions v(k) k=1,2,....

%C To see this, let alpha be the Fibonacci morphism on the alphabet {a,b}:

%C alpha(a) = ab, alpha(b) = a.

%C It is well known that the lower Wythoff sequence u gives the positions of a in the fixed point abaababaab... of alpha, and that the upper Wythoff sequence v gives the positions of b in this infinite Fibonacci word.

%C Let pi be the projection from {1,2,3,4} to {a,b} given by

%C pi(1) = pi(4) = a, pi(2) = pi(3) = b.

%C One easily checks that (composition of morphisms)

%C pi sigma = alpha pi.

%C This implies the remarkable property stated above.

%C What remains to be proved is that lambda(x(u(k)) = a(k) and lambda(x(v(k)) = a(k), where lambda is the letter-to-letter map in the previous comment. We tackle this problem by an extensive analysis of the return words of the letter 1 in the sequence x= 1244343143112...

%C These are the words occurring in x which start with 1 and have no other occurrences of 1 in them.

%C One easily finds that they are given by

%C A:=1, B:=12, C:=124, D:=143, E:=1243, F:=12443, G:=1244343.

%C These words partition x. Moreover, sigma induces a morphism rho on the alphabet {A,B,C,D,E,F,G} given by

%C rho(A) = B, rho(B) =C, rho(C) = F, rho(D) = EA,

%C rho(E) = FA, rho(F) = GA, rho(G) = GDA.

%C CLAIM 1: Let mu be the letter-to-letter map given by

%C mu(A)=mu(B) = 1, mu(C)=mu(D)=mu(E) = 2, mu(F) = 3, mu(G) = 4,

%C then mu(R) = (s(n+1)-s(n)), where R = GDAEABFAB... is the unique fixed point of rho and s = 1,5,7,8,10,... is the sequence of positions of 1 in (x(u(k)).

%C Proof of CLAIM 1: The 1's in x occur exclusively at positions u(k), so if we want the differences s(n+1)-s(n), we can strip the 2's and 3's from the return words of 1, and record how long it takes to the next 1. This is the length of the stripped words A~=1, B~=1, C~=14, ... which is given by mu.

%C CLAIM 2: Let delta be the 'decoration' morphism given by

%C delta(A) = 1, delta(B) = 2, delta(C) = 3, delta(D) = 21,

%C delta(E) = 31, delta(F) = 41, delta(G) = 421,

%C then delta(R) = (t(n+1-t(n)), where rho(R) = R, and t is the sequence of positions of 1 or 3 in the sequence x = 1244343143112....

%C Proof of CLAIM 2: We have to record the differences in the occurrences of 1 and 3 in the return words of 1. These are given by delta. For example: F = 12443, where 1 and 3 occur at position 1 and 5; and the next 1 will occur at the beginning of any of the seven words A,...,G.

%C If we combine CLAIM 1 and CLAIM 2 with lambda(1) = lambda(3) = 0, we obtain the first half of Kimberling's conjecture, simply because delta = mu rho, and rho(R) = R.

%C The second half of the conjecture is obtained in a similar way. (End)

%C From _Michel Dekking_, Mar 10 2020: (Start)

%C The fact that this sequence is a morphic sequence can be easily deduced from the morphism tau given in A007895:

%C tau((j,0)) = (j,0) (j+1,1),

%C tau((j,1)) = (j,0).

%C The sum of digits of a number n in Zeckendorf expansion is given by projecting the n-th letter in the fixed point of tau starting with (0,0) on its first coordinate: (j,i)->j for i=0,1.

%C If we consider all the j's modulo 2, we obtain a morphism sigma' on 4 letters:

%C sigma'((0,0)) = (0,0) (1,1),

%C sigma'((0,1)) = (0,0),

%C sigma'((1,0)) = (1,0)(0,1),

%C sigma'((1,1)) = (1,0).

%C The change of alphabet (0,0)->1, (0,1)->3, (1,0)->4, (1,1)->2, turns sigma' into sigma given in my Nov 28, 2019 comment. The projection map turns into the map 1->0, 2->1, 3->0, 4->1, as in my Nov 28 2019 comment.

%C (End)

%C Another proof that this sequence is a morphic sequence has been given in the monograph by Allouche and Shallit. They demonstrate on page 239 that (a(n)) is a letter to letter projection of a fixed point of a morphism on an alphabet of six letters. However, it can been seen easily that two pairs of letters can be merged, yielding a morphism on an alphabet of four letters, which after a change of alphabet equals the morphism from my previous comments. - _Michel Dekking_, Jun 25 2020

%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, Examples 7.8.2 and 7.8.4.

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 38.11.1, pp. 754-756

%H E. Ferrand, <a href="https://doi.org/10.37236/948">An analogue of the Thue-Morse sequence</a>, The Electronic Journal of Combinatorics, Volume 14 (2007), R30.

%H Leonard Rozendaal, <a href="https://hal.archives-ouvertes.fr/hal-01552281">Pisano word, tesselation, plane-filling fractal</a>, Preprint, hal-01552281, 2017.

%H Jeffrey Shallit, <a href="https://arxiv.org/abs/2010.10956">Subword complexity of the Fibonacci-Thue-Morse sequence: the proof of Dekking's conjecture</a>, arXiv:2010.10956 [cs.DM], 2020.

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F a(n) = A007895(n) mod 2. - _Michel Dekking_, Mar 10 2020

%p A095076 := proc(n)

%p modp(A007895(n),2) ;

%p end proc:

%p seq(A095076(n),n=0..40) ; # _R. J. Mathar_, Sep 22 2020

%t r=(1+5^(1/2))/2; u[n_] := Floor[n*r]; (* A000201 *)

%t a[1] = 0; h = 128;

%t c = (u[#1] &) /@ Range[2h];

%t d = (Complement[Range[Max[#1]], #1] &)[c]; (* A001950 *)

%t Table[a[d[[n]]] = 1 - a[n], {n, 1, h - 1}];

%t Table[a[c[[n]]] = a[n], {n, 1, h}] (* A095076 conjectured *)

%t Flatten[Position[%, 0]] (* A189034 *)

%t Flatten[Position[%%, 1]] (* A189035 *)

%o (Python)

%o def ok(n): return True if n==0 else n*(2*n & n == 0)

%o print([bin(n)[2:].count("1")%2 for n in range(1001) if ok(n)]) # _Indranil Ghosh_, Jun 08 2017

%Y a(n) = A010060(A003714(n)). a(n) = 1 - A095111(n). Characteristic function of A020899. Run counts are given by A095276.

%Y Cf. A007895, A189034, A189035 (positions of 0 and 1 if the conjecture is valid).

%K nonn

%O 0,1

%A _Antti Karttunen_, Jun 01 2004

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Last modified April 11 16:09 EDT 2021. Contains 342886 sequences. (Running on oeis4.)