A342827 -id:A342827 - cp's OEIS frontend

A342818 Length of the longest monochromatic arithmetic progressions of difference d in the Thue-Morse sequence (A010060).

2, 2, 8, 2, 6, 8, 8, 2, 10, 6, 4, 8, 4, 8, 20, 2, 18, 10, 5, 6, 8, 4, 6, 8, 6, 4, 11, 8, 5, 20, 32, 2, 34, 18, 4, 10, 7, 5, 9, 6, 7, 8, 8, 4, 10, 6, 6, 8, 6, 6, 9, 4, 7, 11, 9, 8, 13, 5, 9, 20, 6, 32, 68, 2, 66, 34, 6, 18, 12, 4, 9, 10, 6, 7, 9, 5, 8, 9, 11, 6

Offset: 1

Jeffrey Shallit, Mar 22 2021

nonn

a(n) is the largest l such that there exists k with t(k) = t(k+n) = t(k+2*n) = ... = t(k+(l-1)*n), where t(n) = A010060(n).

			For n = 3, we have t(45) = t(48) = t(51) = t(54) = t(57) = t(60) = t(63) = t(66), and no k and m>8 exist such that t(k) = t(k+3) = t(k+2*3) = ... = t(k+(m-1)*3). So a(3)=8.

Ibai Aedo, Table of n, a(n) for n = 1..2048
Ibai Aedo, Uwe Grimm, Yasushi Nagai, and Petra Staynova, On long arithmetic progressions in binary Morse-like words, arXiv:2101.02056 [math.CO], 2021.
Ibai Aedo, Uwe Grimm, Yasushi Nagai, and Petra Staynova, Monochromatic Arithmetic Progressions in Binary Thue-Morse-Like Words, Theor. Comp. Sci. (2022).
Olga Parshina, On arithmetic index in the generalized Thue-Morse word, arXiv:1811.03884 [math.CO], 2018.

Cf. A010060. Starting position of the first occurrence of a monochromatic arithmetic progression of difference n and length a(n) is given by A342827.

A350226 a(n) is the length of the longest sequence of distinct numbers in arithmetic progression in the interval 0..n, ending with n and where the Thue-Morse sequence (A010060) is constant.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 3, 3, 2, 4, 3, 2, 5, 3, 3, 6, 2, 3, 7, 4, 5, 3, 4, 3, 5, 5, 4, 4, 6, 3, 6, 7, 2, 4, 3, 5, 7, 7, 4, 6, 5, 5, 6, 3, 5, 6, 3, 6, 5, 7, 7, 5, 5, 4, 7, 4, 8, 6, 4, 4, 6, 6, 7, 8, 2, 3, 8, 5, 6, 3, 5, 5, 9, 7, 7, 7, 6, 4, 6, 7, 5, 5, 8, 5, 6, 6, 4

Offset: 0

Rémy Sigrist, Dec 20 2021

nonn

In other words, a(n) is the greatest k > 0 such that A010060(n) = A010060(n - i*d) for i = 0..k-1 and some d > 0 (see A350285 for the least such d).

This sequence is unbounded (this is a consequence of Van der Waerden's theorem).

			For n = 12:
- the first 13 terms of A010060 are:
         0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0
         ^        ^        ^        ^        ^
- A010060(0) = A010060(3) = A010060(6) = A010060(9) = A010060(12),
- and there is no longer sequence of distinct numbers <= 12 in arithmetic progression ending in 12 with this property,
- so a(12) = 5.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, C program for A350226

Cf. A005346, A010060, A342818, A342827, A350235 (records), A350285 (least first differences).

C
```
See Links section.
```

A350600 Color of the first occurrence of a monochromatic arithmetic progression of difference n and length b(n) in the Thue-Morse sequence (A010060), where b(n) = A342818(n).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1

Offset: 1

Ibai Aedo, Jan 08 2022

nonn

Ibai Aedo, Table of n, a(n) for n = 1..2048
Ibai Aedo, Uwe Grimm, Yasushi Nagai, and Petra Staynova, On long arithmetic progressions in binary Morse-like words, arXiv:2101.02056 [math.CO], 2021.
Olga Parshina, On arithmetic index in the generalized Thue-Morse word, arXiv:1811.03884 [math.CO], 2018.

Cf. A010060. A342818, A342827.

a(n) = A010060(A342827(n)).

A370756 a(n) is the van der Waerden number W_t(2,n) of the Thue-Morse word (A010060).

Original entry on oeis.org

1, 3, 7, 10, 13, 16, 19, 57, 73, 136, 151, 166, 181, 196, 211, 226, 241, 256, 271, 621, 652, 683, 714, 745, 776, 807, 838, 869, 900, 931, 962, 993, 1057, 2080, 2143, 2206, 2269, 2332, 2395, 2458, 2521, 2584, 2647, 2710, 2773, 2836, 2899, 2962, 3025, 3088, 3151

Offset: 1

Gandhar Joshi, Feb 29 2024

nonn

a(n) is an extremely naive lower bound of the Waerden numbers A005346(n).

			For n=3, at least a(3)=7 terms of the prefix of the Thue-Morse word are required to find a monochromatic arithmetic progression of length 3:
  Thue-Morse word: 0 1 1 0 1 0 0 ...
                   ^     ^     ^
The 3 terms have equal values and are at locations which are a constant step apart (3 in this case).

B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw. Arch. Wisk. (in German), 15 (1927), 212-216.

Kevin Ryde, Table of n, a(n) for n = 1..8194
Kevin Ryde, C Code

Cf. A010060, A005346, A342818 (longest progression lengths), A342827 (first positions of longest progressions of length A342818(n)).

C
```
/* See links. */
```

// The program is written for a fixed value of progression length, so it is run to find each a(n) separately. Following is an example to find a(5).
def tmw5map "T[i]=T[i+d] & T[i]=T[i+2*d] & T[i]=T[i+3*d] & T[i]=T[i+4*d]";
// This asserts that there is a progression of length 5 for difference d and first position i taken in pair.
def tmw5mapnew "$tmw5map(d,i) & d>0 & i+4*dA342818.
test tmw5mapnew 5;
// This enumerates the first 5 accepted pairs (d,i) in binary listed in lexicographic order. The first or second in the list is our improved bound to be replaced for N in line number 2.
def tmw5mapfin "Ed,i ($tmw5map(d,i) & d>0 & i+4*d

a(13) onward from Kevin Ryde, Mar 31 2024

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A342818 Length of the longest monochromatic arithmetic progressions of difference d in the Thue-Morse sequence (A010060).

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A350226 a(n) is the length of the longest sequence of distinct numbers in arithmetic progression in the interval 0..n, ending with n and where the Thue-Morse sequence (A010060) is constant.

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A350600 Color of the first occurrence of a monochromatic arithmetic progression of difference n and length b(n) in the Thue-Morse sequence (A010060), where b(n) = A342818(n).

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A370756 a(n) is the van der Waerden number W_t(2,n) of the Thue-Morse word (A010060).

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