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The numbers 0, 3, 20 and 119 are the only numbers n under 500 such that for the series S(0) = -n-1; S(1) = n; S(i) = 6*S(i-1) - S(i-2) + 2 the triangular numbers T(S(i)) = S(i) *(S(i) +1)/2 can be factored into R*(R + K) where K is dependent upon n. For n = 0, K is 1: T(-1) = T(0) = 0*1 and...