This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A157199 #18 May 05 2023 18:12:31
%S A157199 9,13,22,26,44,50,25,28
%N 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.
%C A157199 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.
%H A157199 Reed Kelly, <a href="http://www.keldesign.com/math/TCRamsey/Tuple-Chromatic-Ramsey.pdf">On a Generalization of Ramsey Theory</a>, 2009.
%H A157199 Reed Kelly, <a href="http://www.keldesign.com/math/TCRamsey/Code/index.html">Code for Computing Tuple-chromatic Ramsey Numbers and Tuple-chromatic Van der Waerden Numbers</a>
%F A157199 a(r) = tcW(r,r-1,r+1).
%e A157199 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).
%e A157199 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.
%o A157199 (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). */
%Y A157199 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).
%K A157199 nonn,hard,more
%O A157199 2,1
%A A157199 _Reed Kelly_, Feb 25 2009