

A339949


a(n) is the greatest runlength in all nsections of the infinite Fibonacci word A014675.


2



2, 3, 5, 6, 7, 3, 2, 12, 4, 4, 4, 4, 18, 2, 3, 6, 20, 5, 3, 2, 30, 4, 3, 4, 4, 9, 2, 3, 9, 4, 4, 3, 4, 47, 2, 3, 5, 10, 6, 3, 2, 15, 4, 4, 4, 4, 13, 2, 3, 7, 8, 5, 3, 2, 77, 4, 3, 5, 6, 8, 3, 2, 10, 4, 4, 3, 4, 24, 2, 3, 6, 78, 6, 3, 2, 22, 4, 3, 4, 4, 11, 2
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Equivalently a(n) is the greatest runlength in all nsections of the infinite Fibonacci word A003849.
From Jeffrey Shallit, Mar 23 2021: (Start)
We know that the Fibonacci word has exactly n+1 distinct factors of length n.
So to verify a(n) we simply verify there is a monochromatic arithmetic progression of length a(n) and difference n by examining all factors of length (n*a(n)  n + 1) (and we know when we've seen all of them). Next we verify there is no monochromatic AP of length a(n)+1 and difference n by examining all factors of length n*a(n) + 1.
Again, we know when we've seen all of them. (End)


LINKS

Jeffrey Shallit, Table of n, a(n) for n = 1..231
D. Badziahin and J. Shallit, Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences, arXiv:2006.15842 [math.NT], 2020.


EXAMPLE

For n >= 1, r = 0..n, k >= 0, let A014675(n*k+r) denote the kth term of the rth nsection of A014675; i.e.,
(A014675(k)) = 212212122122121221212212212122122121221212212212122121...
has runlengths 1,1,2,1,1,1,2,1,2,1,...; a(1) = 2.
(A014675(2k)) = 22112211222122212221122112221222122211221122112221222...
has runlengths 2,2,2,2,3,1,3,1,3,2,...
(A014675(2k+1)) = 122212221122112211222122211221122112221222122211221...
has runlengths 1,3,1,3,2,2,2,2,2,3,...; a(2) = 3.
(A014675(3k)) = 22111222211122221122222112222211222211122221112222111...
has runlengths 2,3,4,3,4,2,5,2,5,2,4,3,4,3,...
(A014675(3k+1)) = 112222111222211122221112222111222211222221122221112...
has runlengths 2,4,3,4,3,4,3,4,3,4,,5,2,4,3,...
(A014675(3k+2)) = 222211222221122221112222111222211122221112222112222...
has runlengths 4,2,5,2,4,3,4,3,4,3,4,3,4,2,...; a(3) = 5.


MATHEMATICA

r = (1 + Sqrt[5])/2; z = 4000;
f[n_] := Floor[(n + 2) r]  Floor[(n+1) r]; (* A014675 *)
t = Table[Max[Map[Length, Union[Split[Table [f[n m], {n, 0, Floor[z/m]}]]]]], {m, 1, 20}, {n, 1, m}];
Map[Max, t] (* A339949 *)


CROSSREFS

Cf. A001622, A003849, A014675, A339950.
Sequence in context: A023834 A084735 A002734 * A160100 A247891 A354370
Adjacent sequences: A339946 A339947 A339948 * A339950 A339951 A339952


KEYWORD

nonn


AUTHOR

Clark Kimberling, Dec 26 2020


EXTENSIONS

a(61) corrected by Jeffrey Shallit, Mar 23 2021


STATUS

approved



