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A339950 Numbers k such that all k-sections of the infinite Fibonacci word A014675 have just two different run-lengths.

%I A339950 #33 Jan 18 2025 18:25:03
%S A339950 1,7,14,20,27,35,41,48,54,62,69,75,82,90,96,103,109,117,124,130,137,
%T A339950 143,151,158,164,171,179,185,192,198,206,213,219,226,234,240,247,253,
%U A339950 260,268,274,281,287,295,302,308,315,323,329,336,342,350,357,363,370,376,384,391,397,404
%N A339950 Numbers k such that all k-sections of the infinite Fibonacci word A014675 have just two different run-lengths.
%C A339950 Equivalent definition: these are the numbers n such that all n-sections of the infinite Fibonacci word A003849 have just two run-lengths.
%C A339950 The distinct terms of the difference sequence of the first 40 terms are 6, 7, and 8.
%C A339950 Conjecture: a(n) = A189378(n-1)+1 for n >= 2. - _Don Reble_, Apr 06 2021.
%C A339950 "All n-sections" means all subsequences S(k) = (A014675(n*i+k); i = 0, 1, 2, ...), for k = 0, ..., n-1. "Run-lengths" means the numbers of consecutive equal terms in the sequence: see examples. - _M. F. Hasler_, Apr 07 2021
%H A339950 Gandhar Joshi and Dan Rust, <a href="https://arxiv.org/abs/2501.05830">Monochromatic arithmetic progressions in the Fibonacci word</a>, arXiv:2501.05830 [math.DS], 2025. See p. 10.
%e A339950 Let W = A014675, so that as a word, W = 21221212212212122121221221212212212122121221221...
%e A339950 The unique 1-section of W is W itself, which is a concatenation of runs 1, 2, and 22, so that a(1) = 2. The sequence A339949 shows that a(n) > 2 for n = 2,3,4,5,6. For n = 7, the n-section of W that starts with its first letter, 2, is 221221221221221221221221221221221221121..., in which the runs are 22, 1, 11, supporting the conjecture that a(2) = 7.
%e A339950 Some run-lengths may appear quite late. For example, when n = 68, the third run-length appears in the n-section S(k=0) only with the 2829th element, corresponding to the 192372-th element of the original sequence. - _M. F. Hasler_, Apr 07 2021
%t A339950 r = (1 + Sqrt[5])/2; z = 80000;
%t A339950 f[n_] := Floor[(n + 1) r] - Floor[n r];  (* A014675 *)
%t A339950 t = Table[Max[Map[Length,
%t A339950      Union[Split[Table [f[n d], {n, 0, Floor[z/d]}]]]]], {d, 1,
%t A339950     400}, {n, 1, d}];
%t A339950 u = Map[Max, t]
%t A339950 Flatten[Position[u, 2]]  (* A339950 *)
%Y A339950 Cf. A001622, A003849, A014675, A339949.
%Y A339950 See also A189377, A189378, A189379.
%K A339950 nonn
%O A339950 1,2
%A A339950 _Clark Kimberling_, Dec 26 2020
%E A339950 More terms from _Don Reble_, Apr 13 2021