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%I #31 Jan 21 2023 09:09:28
%S 5,3,5,5,3,5,3,5,5,3,5,5,3,5,3,5,5,3,5,3,5,5,3,5,5,3,5,3,5,5,3,5,5,3,
%T 5,3,5,5,3,5,3,5,5,3,5,5,3,5,3,5,5,3,5,3,5,5,3,5,5,3,5,3,5,5,3,5,5,3,
%U 5,3,5,5,3,5,3,5,5,3,5,5,3,5,3,5,5,3
%N Differences of A035336.
%C It appears that this is the Fibonacci word A003849, using 5's and 3's instead of 0's and 1's. In other words, {a(n)} is a fixed point of the morphism 5->53, 3->5.
%C Proof of this conjecture: since A035336(n) = (2*floor(n*phi) + n - 1) (with phi = (1+sqrt(5))/2) is a generalized Beatty sequence, this follows from Lemma 4 in Allouche and Dekking. - _Michel Dekking_, Oct 10 2018
%C Also differences of A089910. - _Bob Selcoe_, Sep 20 2014
%C Proof of this conjecture: this follows from the Carlitz-Scoville-Hoggatt theorem: compositions of the Wythoff A and B sequences are generalized Beatty sequences (cf. Theorem 1 in Allouche and Dekking). - _Michel Dekking_, Oct 10 2018
%H J.-P. Allouche and F. M. Dekking, <a href="https://arxiv.org/abs/1809.03424">Generalized Beatty sequences and complementary triples</a>, arXiv:1809.03424 [math.NT], 2018.
%t Table[2 Floor[n (1 + Sqrt[5])/2] + n - 1, {n, 1, 100}] // Differences (* _Jean-François Alcover_, Dec 14 2018 *)
%Y Cf. A003849, A035336, A089910.
%K nonn
%O 1,1
%A _John W. Layman_, Aug 29 2011
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