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

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

Author

Gandhar Joshi, Feb 29 2024

Keywords

• nonn

Comments

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

Examples

			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).
		

References

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

Links

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

Crossrefs

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

Programs

C

/* See links. */

Walnut

// 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
				

Extensions

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