OFFSET
2,1
COMMENTS
If W(r,k) is the standard multi-color Van der Waerden function with r colors and a required monochrome arithmetic subsequence of length k, then tcW(r,1,k) = W(r,k). In tcW(r,1,k), the 1 would indicate a monochrome subsequence. For tcW(r,2,k) an arithmetic subsequence of length k in 1 OR 2 colors would match the criteria. For tcW(r,3,k) an arithmetic subsequence of length k in 1, 2, or 3 colors suffices.
LINKS
FORMULA
a(r) = tcW(r,r-1,r+1).
EXAMPLE
a(2) = tcW(2,1,3) = W(2,3) = 9. If {1,...,9} is colored in 2 colors, then a 3-term arithmetic subsequence exists in 1 color (monochrome).
a(3) = tcW(3,2,4) = 13. If {1,...,13} is colored in 3 colors, then a 4-term arithmetic subsequence exists in at most 2 colors.
PROG
(C++) /* A C++ program is available from the links. It is not the best program, but it is relatively fast. To get the terms of the above sequence, you have to compile the program and choose parameters such as: find_vdw 10000 5 4 6 for tcW(5, 4, 6) and find_vdw 10000 6 5 7 for tcW(6, 5, 7). */
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Reed Kelly, Feb 25 2009
STATUS
approved