15 research outputs found
On -extensions of the Hankel determinants of certain automatic sequences
In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morse
sequence, and proved that all the Hankel determinants of the period-doubling
sequence are odd integral numbers. We speak of -extension when the entries
along the diagonal in the Hankel determinant are all multiplied by~. Then we
prove that the -extension of each Hankel determinant of the period-doubling
sequence is a polynomial in , whose leading coefficient is the {\it only
one} to be an odd integral number. Our proof makes use of the combinatorial
set-up developed by Bugeaud and Han, which appears to be very suitable for this
study, as the parameter counts the number of fixed points of a permutation.
Finally, we prove that all the -extensions of the Hankel determinants of the
regular paperfolding sequence are polynomials in of degree less than or
equal to
Hankel Determinant Calculus for the Thue-Morse and related sequences
The Hankel determinants of certain automatic sequences are evaluated,
based on a calculation modulo a prime number. In most cases, the Hankel
determinants of automatic sequences do not have any closed-form expressions;
the traditional methods, such as -decompo\-si\-tion and Jacobi continued
fraction, cannot be applied directly. Our method is based on a simple idea: the
Hankel determinants of each sequence equal to modulo are equal to
the Hankel determinants of modulo . The clue then consists of finding a
nice sequence , whose Hankel determinants have closed-form expressions.
Several examples are presented, including a result saying that the Hankel
determinants of the Thue-Morse sequence are nonzero, first proved by Allouche,
Peyri\`ere, Wen and Wen using determinant manipulation. The present approach
shortens the proof of the latter result significantly. We also prove that the
corresponding Hankel determinants do not vanish when the powers in the
infinite product defining the Thue--Morse sequence are replaced by
The Pagoda Sequence: a Ramble through Linear Complexity, Number Walls, D0L Sequences, Finite State Automata, and Aperiodic Tilings
We review the concept of the number wall as an alternative to the traditional
linear complexity profile (LCP), and sketch the relationship to other topics
such as linear feedback shift-register (LFSR) and context-free Lindenmayer
(D0L) sequences. A remarkable ternary analogue of the Thue-Morse sequence is
introduced having deficiency 2 modulo 3, and this property verified via the
re-interpretation of the number wall as an aperiodic plane tiling