

A121385


Minimal number of monochromatic threeterm arithmetic progressions that a twocoloring 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
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OFFSET

1,11


COMMENTS

a(9) = 1 is the wellknown fact that the van der Waerden number for two colors and threeterm arithmetic progressions is 9.
From Rob Pratt, May 27 2014: (Start)
By ignoring the last element, we see that a(n) >= a(n1).
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+(k1)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 k1} 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

Table of n, a(n) for n=1..45.


EXAMPLE

a(8) = 0 because we can two color {1,...,8} by 11001100 so that there are no monochromatic threeterm 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



