On the regularity of the Hankel determinant sequence of the characteristic sequence of powers of 2

Abstract

For any sequences u = {u(n)} n≥0 , v = {v(n)} n≥0 , we define uv := {u(n)v(n)} n≥0 and u + v := {u(n) + v(n)} n≥0 . Let f i (x) (0 ≤ i < k) be sequence polynomials whose coefficients are integer sequences. We say an integer sequence u = {u(n)} n≥0 is a polynomial generated sequence ifIn this paper, we study the polynomial generated sequences. Assume k ≥ 2 and f i (x) = a i x+b i (0 ≤ i < k). If a i are k-automatic and b i are k-regular for 0 ≤ i < k, then we prove that the corresponding polynomial generated sequences are k-regular. As a application, we prove that the Hankel determinant sequence {det(p(i+j)) n−1 i,j=0 } n≥0 is 2-regular, where {p(n)} n≥0 = 0110100010000 · · · is the characteristic sequence of powers 2. Moreover, we give a answer of Cigler's conjecture about the Hankel determinants.