Dyadic Expansion of the Apwenian Recursion

Abstract

Iterating the apwenian recursion expresses each entry as the sum over a consecutive interval of binary descendants.

Theorem 1.1 (Sum over binary descendants).

Lean statement: D5/S3/Combinatorics/Apwenian/GuoHanDyadic.dyadic_expansion

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Apwenian/GuoHanDyadic.dyadic_expansion (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Ying-Jun Guo, Guo-Niu Han (2025). On a family of automatic apwenian sequences. DOI: 10.1016/j.disc.2025.114399. URL: https://irma.math.unistra.fr/~guoniu/papers/p120autoapw.pdf.

Commentary.

For every sequence b in the integers modulo two satisfying b(n) = b(2n + 1) + b(2n + 2) for all nonnegative n, and every pair of nonnegative integers h and n, b(n) is the sum of b(2^h(n + 1) - 1 + j) over j from zero through 2^h minus one. The sum is taken modulo two. At depth zero it consists of b(n) alone; at each successive depth the two descendants of each term partition the next consecutive interval.

References

• Truth anchor: D5/S3/Combinatorics/Apwenian/GuoHanDyadic.dyadic_expansion
• Dependency: D5/S3/Combinatorics/Apwenian/GuoHanBlocks