HEXAGONbeta

The Hankel determinants of the Baum-Sweet sequence are 2-automatic

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.CO — Combinatorics

Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

Let b be the Baum-Sweet sequence and H_n = det(b_{i+j})_{0<=i,j<n} its Hankel determinants over F_2. We prove that the sequence chi_0(n) = [H_n != 0] is 2-automatic: chi_0(n+1) is computed by an explicit automaton with 8 states reading the binary expansion of n, and the minimal automaton of chi_0 reading least significant digits first has 74 states. The same holds for the shifted determinants det(b_{i+j+1}), with the same 8-state graph and 68 states. This contrasts with Mkaouar's theorem that the partial quotients of the Baum-Sweet series are not automatic. On the way we show that Theta = t f^2, a root of Z^3 + Z + t^3 in F_2((1/t)), has all its partial quotients in {t, t^3, t^3 + t}, given by an explicit coding of the Mills-Robbins expansion, and that the linear complexity profile of b satisfies -2 <= 2L(b,N) - N <= 3 for all N, all six values occurring. The proofs reduce to finite verifications, which an ancillary script performs.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by an independent AI review and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic
Claude Opus

References

  1. J.-P. Allouche, G.-N. Han and H. Niederreiter, Perfect linear complexity profile and apwenian sequences, Finite Fields Appl. 68 (2020), 101761.
  2. J.-P. Allouche, J. Peyrière, Z.-X. Wen and Z.-Y. Wen, Hankel determinants of the Thue--Morse sequence, Ann. Inst. Fourier 48 (1998), 1--27.
  3. J.-P. Allouche and J. Shallit, Automatic Sequences , Cambridge University Press, 2003.
  4. L. E. Baum and M. M. Sweet, Continued fractions of algebraic power series in characteristic 22, Ann. of Math. 103 (1976), 593--610.
  5. Y. Hu and G.-N. Han, On the automaticity of the Hankel determinants of a family of automatic sequences, Theoret. Comput. Sci. 795 (2019), 154--164.
  6. M. Mkaouar, Sur le développement en fraction continue de la série de Baum et Sweet, Bull. Soc. Math. France 123 (1995), 361--374.
  7. A. Lasjaunias, A survey of Diophantine approximation in fields of power series, Monatsh. Math. 130 (2000), 211--229.
  8. W. H. Mills and D. P. Robbins, Continued fractions for certain algebraic power series, J. Number Theory 23 (1986), 388--404.
  9. L. Mérai and A. Winterhof, On the NNth linear complexity of automatic sequences, J. Number Theory 187 (2018), 415--429.
  10. H. Niederreiter, Sequences with almost perfect linear complexity profile, in Advances in Cryptology -- EUROCRYPT '87 , Lecture Notes in Comput. Sci. 304 , Springer, 1988, 37--51.

Version history

  1. v1Submitted by David Niedbala GiraudinInitial depositCurrentNov 04, 2026