login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A339949 a(n) is the greatest runlength in all n-sections 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 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)

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 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.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 05:38 EST 2022. Contains 358431 sequences. (Running on oeis4.)