A370755 a(n) is the van der Waerden number W_f(2,n) of the Fibonacci word (A003849).

1, 3, 8, 12, 21, 29, 42, 59, 67, 80, 88, 144, 160, 173, 186, 199, 220, 254, 377, 394, 423, 444, 465, 491, 512, 533, 554, 588, 609, 987, 1024, 1058, 1092, 1126, 1160, 1194, 1228, 1262, 1296, 1330, 1364, 1406, 1440, 1474, 1508, 1563, 1652, 2588, 2643, 2698, 2753

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)=8 terms of the prefix of the Fibonacci word are required to find a monochromatic arithmetic progression of length 3:
  Fibonacci word: 0 1 0 0 1 0 1 0 ...
                        ^   ^   ^
The 3 terms have equal values and are at locations which are a constant step apart (2 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..7500
• Kevin Ryde, C Code

Crossrefs

Cf. A003849, A005346, A339949 (longest progression lengths), A364648 (first positions of longest progressions of length A339949(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 fibw5map "?msd_fib F[i]=F[i+d] & F[i]=F[i+2*d] & F[i]=F[i+3*d] & F[i]=F[i+4*d]";
// This asserts that there is a progression of length 5 for difference d and first position i taken in pair.
def fibw5mapnew "?msd_fib $fibw5map(d,i) & d>0 & i+4*dA339949.
test fibw5mapnew 5;
// This enumerates the first 5 accepted pairs (d,i) in Zeckendorf representation listed in lexicographic order. The first or second in the list is our improved bound to be replaced for N in line number 2.
def fibw5mapfin "?msd_fib Ed,i ($fibw5map(d,i) & d>0 & i+4*d