OFFSET
1,2
COMMENTS
a(n) is an extremely naive lower bound of the Waerden numbers A005346(n).
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
EXAMPLE
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).
PROG
(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 $fibw5map(d, i) & d>0 & i+4*d<N";
// This accepts 2-tuple (d, i) such that the last progression appears before N. In the code, N must be replaced with an integer value. We take a calculated guess of what N=i+(n-1)*d is from the list of longest progression lengths A339949.
test fibw5mapnew 5;
// This enumerates the first 5 accepted pairs (d, i) in Zeckendorf representation listed in lexicographic order. The first or second in the list is our improved bound to be replaced for N in line number 2.
def fibw5mapfin "?msd_fib Ed, i ($fibw5map(d, i) & d>0 & i+4*d<N')";
// This checks if there is any pair (d, i) such that progression length 5 appears before N' which is our improved bound. If Walnut outputs FALSE, then a(n)=N'+1.
(C) /* See links. */
CROSSREFS
KEYWORD
nonn
AUTHOR
Gandhar Joshi, Feb 29 2024
STATUS
approved