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Let gj denote the largest integer that is represented exactly j times as a non-negative integer linear combination of { x1, . . . , xn}. We show that for any k > 0, and n == 5, the quantity g0 -- gk is unbounded. Furthermore, we provide examples with g0 > gk for n ≥≥ 6 and g0 > g1 for n ≥≥ 4. [ABSTRACT FROM AUTHOR]

Independently, Pirillo and Varricchio, Halbeisen and Hungerbüühler and Freedman considered the following problem, open since 1992: Does there exist an infinite word w over a finite subset of ℤℤ such that w contains no two consecutive blocks of the same length and sum? We consider some variations on this problem in the light of van der Waerden's theorem on arithmetic progressions. [ABSTRACT FROM AUTHOR]