Hankel Matrices for the Period-Doubling Sequence

@article{Fokkink2015HankelMF,
  title={Hankel Matrices for the Period-Doubling Sequence},
  author={Robbert J. Fokkink and Cor Kraaikamp and Jeffrey Shallit},
  journal={ArXiv},
  year={2015},
  volume={abs/1511.06569}
}
We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders. 
Some observations on the Rueppel sequence and associated Hankel determinants
Starting with a definition based on the Catalan numbers, we carry out an empirical study of the Rueppel sequence. We use the Hankel transform as the main technique. By means of this transform we findExpand
On the automaticity of the Hankel determinants of a family of automatic sequences
TLDR
A partial answer to the question of the automaticity of the reduced Hankel determinants modulo $2$ of $\pm 1$-automatic sequences with kernel of cardinality at most $2$. Expand
On the automaticity of sequences defined by the Thue–Morse and period-doubling Stieltjes continued fractions
Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partialExpand
Hankel determinants of a Sturmian sequence.
Let $\tau$ be the substitution $1\to 101$ and $0\to 1$ on the alphabet $\{0,1\}$. The fixed point of $\tau$ leading by 1, denoted by $\mathbf{s}$, is a Sturmian sequence. We first give aExpand

References

SHOWING 1-10 OF 32 REFERENCES
Hankel determinants of the Cantor sequence
In the paper, we give the recurrent equations of the Hankel determinants of the Cantor sequence, and show that the Hankel determinants as a double sequence is 3-automatic. With the help of the HankelExpand
A Combinatorial Proof of the Non-Vanishing of Hankel Determinants of the Thue-Morse Sequence
TLDR
An alternative, purely combinatorial proof of the Hankel determinants associated with the Thue-Morse sequence are always nonzero is presented and a recent result of Coons is re-proved on the non-vanishing of the Hankel determinant associated to two other classical integer sequences. Expand
Local symmetries in the period-doubling sequence
  • D. Damanik
  • Computer Science, Mathematics
  • Discret. Appl. Math.
  • 2000
TLDR
The number p ( n ) of palindromes oflength n and the number b k ( n) of k th powers of words of length n occurring in this sequence are computed explicitly. Expand
Automaticity of the Hankel determinants of difference sequences of the Thue-Morse sequence
TLDR
It is proved that for any k, the two-dimensional sequence (modulo 2) {|Δk(t)np|}n,p,k≥0 is 2-automatic. Expand
On t-extensions of the Hankel determinants of certain automatic sequences
TLDR
It is proved that the t-extension of each Hankel determinant of the period-doubling sequence is a polynomial in t, whose leading coefficient is the only one to be an odd integer. Expand
Evaluations of the Hankel determinants of a Thue–Morse-like sequence
We obtain simple relations between the Hankel determinants of the formal power series Q k≥0 (1 + Jx 3 k) where J = (√ −3 − 1)/2, and prove that the sequence of Hankel determinants is an aperiodicExpand
On the irrationality exponent of the regular paperfolding numbers
Abstract In this paper, improving the method of Allouche et al., we calculate the Hankel determinant of the regular paperfolding sequence, and prove that the Hankel determinant sequence modulo 2 isExpand
On rational approximations of certain Mahler functions with a connection to the Thue–Morse sequence
We shall obtain the irrationality exponent 2 for some values of two special Mahler functions. This gives a new proof for the recent result of Bugeaud on Thue–Morse–Mahler numbers.
Hankel Determinant Calculus for the Thue-Morse and related sequences
The Hankel determinants of certain automatic sequences $f$ are evaluated, based on a calculation modulo a prime number. In most cases, the Hankel determinants of automatic sequences do not have anyExpand
Computer assisted proof for Apwenian sequences related to Hankel determinants
An infinite ±1-sequence is called Apwenian if its Hankel determinant of order n divided by 2 n−1 is an odd number for every positive integer n. In 1998, Allouche,Peyri ere, Wen and Wen discovered andExpand
...
1
2
3
4
...