Numeration Systems and Fractal Sequences

Let N denote the set of positive integers. Every sequence B = (b0, b1, . . .) of numbers in N satisfying (1) 1 = b0 < b1 < . . . is a basis for N, as each n in N has a B-representation (2) n = c0b0 + c1b1 + . . .+ ckbk, where bk ≤ n < bk+1 and the coefficients ci are given by the division algorithm: (3) n = ckbk + rk, ck = [n/bk], 0 ≤ rk < bk and (4) ri = ci−1bi−1 + ri−1, ci−1 = [ri/bi−1], 0 ≤ ri−1 < bi−1 for 1 ≤ i < k. In (2) let i be the least index h such that ch 6= 0; then bi is the B-residue of n. A proper basis is a basis other than the sequence (1, 2, . . .) consisting of all the positive integers. We extend the above notions to finite sequences Bj = (b0, b1, . . . , bj) satisfying 1 = b0 < b1 < . . . < bj for j ≥ 0. Such a finite sequence is a finite basis, and a Bj-representation is a sum (2) c0b0 + c1b1 + . . . + cjbj such that if n = c0b0+c1b1+ . . .+cjbj , then there exist integers r0, r1, . . . , rj such that (3) n = cjbj + rj , cj = [n/bj−1], 0 ≤ rj < bj