A157199 The terms of this sequence are the first several terms of tcW(r,r-1,r+1), where r=2,3,4,.... Informally, the function tcW is like the multi-color Van der Waerden function W, except that the second parameter determines the number of colors found in the target subsequence. See links for definition.

9, 13, 22, 26, 44, 50, 25, 28

Offset: 2

Author

Reed Kelly, Feb 25 2009

Keywords

• nonn
• hard
• more

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.

Examples

			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.
		

Links

• Reed Kelly, On a Generalization of Ramsey Theory, 2009.
• Reed Kelly, Code for Computing Tuple-chromatic Ramsey Numbers and Tuple-chromatic Van der Waerden Numbers

Crossrefs

Another part of the tcW function: A157102. The 2-color Van der Waerden numbers: A005346, W(2, k). Multi-color Van der Waerden numbers with 3-term monochrome arithmetic subsequences A135415, W(r, 3).

Formula

a(r) = tcW(r,r-1,r+1).