Alternatingbase

Alternatingbase is a numeral system in which the radix (base) used for each digit position follows a fixed repeating pattern, rather than being the same across all positions. Formally, let b1, b2, ..., bk be integers greater than 1 describing a periodic base pattern of length k. A number is represented by digits d0, d1, ..., dm-1 with 0 ≤ di < b_{i mod k}. The value of the representation is the sum d0 + d1·b1 + d2·(b1·b2) + d3·(b1·b2·b3) + ...; equivalently, each position multiplies the accumulated product of the previous bases.

Conversion between alternatingbase representations and integers is a standard mixed-radix procedure. To encode N, repeatedly divide by the current base in the pattern: di = N mod b_{i mod k}, N := floor(N / b_{i mod k}); continue for as many digits as needed. To decode, reconstruct N by summing di multiplied by the appropriate product of prior bases. For a fixed length m, the total number of representable values is the product of the bases over the first m positions.

Example: using the pattern [2, 3] repeatedly, the digits (d0, d1, d2) with d0 ∈ {0,1}, d1 ∈ {0,1,2}, d2 ∈ {0,1} encode values up to 11. The value is d0 + 2·d1 + 6·d2.

Alternatingbase is a special case of a mixed-radix numeral system. If all bases are equal, it reduces to ordinary base-B representation; if the bases increase or follow other nonperiodic sequences, it aligns with broader mixed-radix concepts. See also mixed radix numeral systems, factorial number system, and positional numeral systems.