183pattern

183pattern is a class of numeric sequences described in puzzle discussions and educational materials as those formed by a fixed cycle of increments consisting of 1, 8, and 3. The pattern is named for the sequence of increments rather than the resulting terms, and it illustrates how simple cyclical rules can generate nontrivial growth.

Construction and example: Choose an initial value a0. Then for n ≥ 0, define a_{n+1} = a_n + d_{n mod 3} where d = [1, 8, 3]. For instance, starting with a0 = 0, the first several terms are 0, 1, 9, 12, 13, 21, 24, 25, 33, 36, 37, 45, and so on. After every three steps the total increase is 12, so a_{n+3} = a_n + 12. This yields a repeating residue pattern modulo 12 and a predictable long-term growth.

Properties: A key feature is that the three-step cycle sums to 12, making the sequence progress in a quasi-arithmetic fashion with a periodic residue cycle modulo 12. The construction demonstrates how initial conditions and a small set of rules control the trajectory of a sequence over time.

Variants and usage: Similar patterns can be formed with different cyclic increments, such as [2, 5, 7] or other triplets, producing analogous structural properties. The term 183pattern is primarily used in recreational mathematics, puzzle design, and classroom exercises to explore recurrence relations, modular arithmetic, and sequence analysis rather than in formal mathematical theory.

History and see also: The term 183pattern appears in informal online discussions and educational blogs without a single canonical publication describing it. See also arithmetic progression, modular arithmetic, recurrence relation, and integer sequences.