Statement

Giorgio is now studying the interesting codes, that is, sequences of N digits 0 or 1 such that for any possible ratio x = 1, …, 10 and starting value i = 0, …, N-3x-1, the digits in the positions given by the corresponding arithmetic progression (i,i+x,i+2x,i+3x) are not all equal to each other. For example, the following 10-digit codes are interesting: While these are not: because of the arithmetic progressions (i,x) respectively (0,3), (1,1), (2,2). How many interesting codes with N digits and containing exactly K digits equal to 1 there exist?