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 A121385 Minimal number of monochromatic three-term arithmetic progressions that a two-coloring of {1,...,n} can contain. 1
 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 62, 66, 70, 74, 78, 82, 86 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,11 COMMENTS a(9) = 1 is the well-known fact that the van der Waerden number for two colors and three-term arithmetic progressions is 9. From Rob Pratt, May 27 2014: (Start) By ignoring the last element, we see that a(n) >= a(n-1). The general problem for k terms and r colors can be solved by using integer programming, with binary decision variables x[i,c] = 1 if element i receives color c and 0 otherwise, y[i,d] = 1 if arithmetic progression (i,i+d,...,i+(k-1)d) is monochromatic and 0 otherwise. The constraints are sum {c in 1..r} x[i,c] = 1 for all i in 1, …, n and sum {j=0 to k-1} x[i+j*d,c] - k + 1 <= y[i,d] for all i, d, c.  The objective is to minimize sum {i, d} y[i,d]. Upper bounds are a(46) <= 90, a(47) <= 95, a(48) <= 100, a(49) <= 104, a(50) <= 110. (End) LINKS EXAMPLE a(8) = 0 because we can two color {1,...,8} by 11001100 so that there are no monochromatic three-term arithmetic progressions. CROSSREFS Cf. A005346, A121386. Sequence in context: A017874 A029016 A290807 * A029015 A000008 A001312 Adjacent sequences:  A121382 A121383 A121384 * A121386 A121387 A121388 KEYWORD nonn,more AUTHOR Steve Butler, Jul 26 2006 EXTENSIONS a(35)-a(45) from Rob Pratt, May 27 2014 STATUS approved

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Last modified May 25 01:00 EDT 2022. Contains 354047 sequences. (Running on oeis4.)