Previously: Anyhow... the prime question would seem to be: What is the minimum number of constants you need to know, to be able to know all of a[1], a[2],..., a[N] since they all are just integer linear combinations of members of your master set? Certainly if N>0 the answer is at most ceiling((N+1)/2), from the Gosper-Hunt relations; and all of my relations for a[] listed above only determine a[2k]+a[2k+1] and not a[2k+1] alone, which suggests perhaps the Gosper-Hunt relations might be a full basis for the relations obeyed by a[]. But now: If we just focus on these relations: 1*a[1]-3*a[2]-2*a[3] = -0.0000000000000000 1*a[1]-5*a[2]+4*a[4]+4*a[5] = -0.0000000000000000 1*a[1]-5*a[2]+8*a[4]-8*a[8]-8*a[9] = -0.0000000000000000 1*a[2]-3*a[4]-2*a[5]-2*a[6]-2*a[7] = -0.0000000000000000 1*a[3]-1*a[4]-2*a[6]-2*a[7] = 0.0000000000000000 1*a[3]-1*a[4]-4*a[6]+4*a[12]+4*a[13] = 0.0000000000000000 1*a[3]-1*a[4]-4*a[7]-4*a[12]-4*a[13] = 0.0000000000000000 1*a[4]-1*a[5]-2*a[8]-2*a[9] = 0.0000000000000000 1*a[4]-1*a[5]-4*a[8]+4*a[16]+4*a[17] = 0.0000000000000000 1*a[4]-1*a[5]-4*a[9]-4*a[16]-4*a[17] = 0.0000000000000000 1*a[5]-1*a[6]-2*a[10]-2*a[11] = 0.0000000000000000 1*a[5]-1*a[6]-4*a[10]+4*a[20]+4*a[21] = 0.0000000000000000 1*a[5]-1*a[6]-4*a[11]-4*a[20]-4*a[21] = 0.0000000000000000 1*a[6]-1*a[7]-4*a[12]+4*a[24]+4*a[25] = -0.0000000000000000 1*a[6]-1*a[7]-4*a[13]-4*a[24]-4*a[25] = -0.0000000000000000 we see that only 5 constants a1, a2, a16+a17, a20+a21, a24+a25 suffice to determine a1,a2,...,a13. Perhaps the answer is you need N constants to determine a[1],a[2],...,a[2N+3]... but I think this is still compatible with the hypothesis that the Gosper-Hunt relations are the full basis. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)