

A157199


See links for definition. Specifically, the terms of this sequence are the first several terms of tcW(r,r1,r+1), where r=2,3,4.... Informally, the function tcW is like the multicolor Van der Waerden function W, except that the second parameter determines the number of colors found in the target subsequence. If W(r,k) is the standard multicolor 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.


0




OFFSET

2,1


COMMENTS

a(r) = tcW(r,r1,r+1)


LINKS

Table of n, a(n) for n=2..9.
Reed Kelly, http://www.keldesign.com/math/TCRamsey/TupleChromaticRamsey.pdf
Reed Kelly, http://www.keldesign.com/math/TCRamsey/Code/index.html


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.


MAPLE

(Other) 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

Another part of the tcW function: A157102. The 2color Van der Waerden numbers: A005346, W(2, k). Multicolor Van der Waerden numbers with 3 term monochrome arithmetic subsequences A135415, W(r, 3).
Sequence in context: A137168 A033870 A151901 * A242919 A097539 A107913
Adjacent sequences: A157196 A157197 A157198 * A157200 A157201 A157202


KEYWORD

hard,nonn


AUTHOR

Reed Kelly, Feb 25 2009


STATUS

approved



