

A339950


Numbers k such that all ksections of the infinite Fibonacci word A014675 have just two different runlengths.


3



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
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OFFSET

1,2


COMMENTS

Equivalent definition: these are the numbers n such that all nsections of the infinite Fibonacci word A003849 have just two runlengths.
The distinct terms of the difference sequence of the first 40 terms are 6, 7, and 8.
Conjecture: a(n) = A189378(n1)+1 for n >= 2.  Don Reble, Apr 06 2021.
"All nsections" means all subsequences S(k) = (A014675(n*i+k); i = 0, 1, 2, ...), for k = 0, ..., n1. "Runlengths" means the numbers of consecutive equal terms in the sequence: see examples.  M. F. Hasler, Apr 07 2021


LINKS

Table of n, a(n) for n=1..60.


EXAMPLE

Let W = A014675, so that as a word, W = 21221212212212122121221221212212212122121221221...
The unique 1section 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 nsection 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 runlengths may appear quite late. For example, when n = 68, the third runlength appears in the nsection S(k=0) only with the 2829th element, corresponding to the 192372th element of the original sequence.  M. F. Hasler, Apr 07 2021


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 *)


CROSSREFS

Cf. A001622, A003849, A014675, A339949.
See also A189377, A189378, A189379.
Sequence in context: A246393 A246305 A308015 * A242888 A037367 A276613
Adjacent sequences: A339947 A339948 A339949 * A339951 A339952 A339953


KEYWORD

nonn


AUTHOR

Clark Kimberling, Dec 26 2020


EXTENSIONS

More terms from Don Reble, Apr 13 2021


STATUS

approved



