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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)