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

1, 7, 14, 20, 27, 35, 41, 48, 54, 62, 69, 75, 82, 90, 96, 103, 109, 117, 124, 130, 137, 143, 151, 158, 164, 171, 179, 185, 192, 198, 206, 213, 219, 226, 234, 240, 247, 253, 260, 268, 274, 281, 287, 295, 302, 308, 315, 323, 329, 336, 342, 350, 357, 363, 370, 376, 384, 391, 397, 404

Offset: 1

Author

Clark Kimberling, Dec 26 2020

Keywords

• nonn

Comments

Equivalent definition: these are the numbers n such that all n-sections of the infinite Fibonacci word A003849 have just two run-lengths.

The distinct terms of the difference sequence of the first 40 terms are 6, 7, and 8.

Conjecture: a(n) = A189378(n-1)+1 for n >= 2. - Don Reble, Apr 06 2021.

"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

Examples

			Let W = A014675, so that as a word, W = 21221212212212122121221221212212212122121221221...
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.
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
		

Links

• Gandhar Joshi and Dan Rust, Monochromatic arithmetic progressions in the Fibonacci word, arXiv:2501.05830 [math.DS], 2025. See p. 10.

Crossrefs

Cf. A001622, A003849, A014675, A339949.

See also A189377, A189378, A189379.

Programs

Mathematica

r = (1 + Sqrt[5])/2; z = 80000;
f[n_] := Floor[(n + 1) r] - Floor[n r];  (* A014675 *)
t = Table[Max[Map[Length,
     Union[Split[Table [f[n d], {n, 0, Floor[z/d]}]]]]], {d, 1,
    400}, {n, 1, d}];
u = Map[Max, t]
Flatten[Position[u, 2]]  (* A339950 *)

Extensions

More terms from Don Reble, Apr 13 2021