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 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

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