A339949 a(n) is the greatest runlength in all n-sections of the infinite Fibonacci word A014675.

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

Offset: 1

Author

Clark Kimberling, Dec 26 2020

Keywords

• nonn
• easy

Comments

Equivalently a(n) is the greatest runlength in all n-sections 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)

Examples

			For n  >= 1,  r =  0..n, k >= 0, let A014675(n*k+r) denote the k-th term of the r-th n-section 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.
		

Links

• Gandhar Joshi, Table of n, a(n) for n = 1..10000 (terms 1..232 from Jeffrey Shallit).
• Dmitry Badziahin and Jeffrey Shallit, Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences, arXiv:2006.15842 [math.NT], 2020.
• Gandhar Joshi and Dan Rust, Monochromatic arithmetic progressions in the Fibonacci word, arXiv:2501.05830 [math.NT], 2025. See p. 9.

Crossrefs

Cf. A001622, A003849, A014675, A339950.

Programs

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 *)

PARI

phi = quadgen(5);
g(n) = min(frac(n * phi), 1 - frac(n * phi));
a(n) = if (g(n) <= (1 / phi)^2, ceil((1 / phi) / g(n)), 2*ceil(((1 / phi) - g(n)) / g(2 * n))); \\ Gandhar Joshi, Jan 14 2025

Formula

From Gandhar Joshi, Jan 14 2025: (Start)

phi = the golden ratio. g(n) = min {n*phi mod 1, 1 - (n*phi mod 1)}.

If g(n) <= (phi)^(-2), a(n) = ceiling{((phi)^(-1))/g(n)};

otherwise, a(n) = 2*ceiling{((phi)^(-1)-g(n))/g(2n)}. (End)

Extensions

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