A265025 Determinants of the Hankel matrices for the period-doubling sequence A035263.

1, 1, -1, -3, 1, 1, -1, -15, 1, 1, -1, -3, 1, 1, -9, -495, 9, 1, -1, -3, 1, 1, -1, -15, 1, 1, -1, -3, 9, 81, -2025, -467775, 2025, 81, -9, -3, 1, 1, -1, -15, 1, 1, -1, -3, 1, 1, -9, -495, 9, 1, -1, -3, 1, 1, -1, -15, 9, 81, -729, -19683, 164025, 4100625, -496175625, -448046589375, 496175625, 4100625, -164025, -19683, 729, 81

Offset

1,4

Comments

The n-th Hankel matrix of the sequence is formed by making an n X n matrix with each row a successive length-n "window" into the sequence.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Robbert Fokkink, Cor Kraaikamp, and Jeffrey Shallit, Hankel matrices for the period-doubling sequence, arxiv preprint arXiv:1511.06569 [math.CO], 2015-2016.

Formula

a(2^k) = (-1)*A001045(k+1)*Product_{i=0..k-3} A001045(k-i)^(2^i) for k>=3.

Mathematica

periodDouble[n_] :=Module[{A = {0, 1}}, For[i = 2, i <= n, i++, AppendTo[A, If[EvenQ[i], 1 - A[[ Floor[i/2] ]], 1]]]; A];
a[n_] := Module[{A, M}, A = periodDouble[2n-1]; M = Table[If[i == 0, 1, A[[i]]] , {j, 0, n-1}, {i, j, n+j-1}]; Det[M]];
Array[a, 70] (* Jean-François Alcover, Aug 12 2018, after Tom Edgar *)

Prog

(Sage)
def periodDouble(n):
    A=[0, 1]
    for i in [2..n]:
        if i%2==0:
            A.append(1-A[floor(i/2)])
        else:
            A.append(1)
    return A[1:]
def a(n):
    A=periodDouble(2*n-1)
    M=matrix([[A[i] for i in [j..n+j-1]] for j in [0..n-1]])
    return det(M)
[a(i) for i in [1..70]] # Tom Edgar, Nov 30 2015

Crossrefs

Cf. A035263, A001045.

Sequence in context: A331568 A010278 A137795 * A284578 A070989 A274741

Adjacent sequences: A265022 A265023 A265024 * A265026 A265027 A265028

Keyword

sign,look

Author

Jeffrey Shallit, Nov 30 2015

Status

approved