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%I A265025 #24 Aug 13 2018 09:08:11
%S A265025 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,
%T A265025 -1,-3,9,81,-2025,-467775,2025,81,-9,-3,1,1,-1,-15,1,1,-1,-3,1,1,-9,
%U A265025 -495,9,1,-1,-3,1,1,-1,-15,9,81,-729,-19683,164025,4100625,-496175625,-448046589375,496175625,4100625,-164025,-19683,729,81
%N A265025 Determinants of the Hankel matrices for the period-doubling sequence A035263.
%C A265025 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.
%H A265025 Alois P. Heinz, <a href="/A265025/b265025.txt">Table of n, a(n) for n = 1..1000</a>
%H A265025 Robbert Fokkink, Cor Kraaikamp, and Jeffrey Shallit, <a href="http://arxiv.org/abs/1511.06569">Hankel matrices for the period-doubling sequence</a>, arxiv preprint  arXiv:1511.06569 [math.CO], 2015-2016.
%F A265025 a(2^k) = (-1)*A001045(k+1)*Product_{i=0..k-3} A001045(k-i)^(2^i) for k>=3.
%t A265025 periodDouble[n_] :=Module[{A = {0, 1}}, For[i = 2, i <= n, i++, AppendTo[A, If[EvenQ[i], 1 - A[[ Floor[i/2] ]], 1]]]; A];
%t A265025 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]];
%t A265025 Array[a, 70] (* _Jean-François Alcover_, Aug 12 2018, after _Tom Edgar_ *)
%o A265025 (Sage)
%o A265025 def periodDouble(n):
%o A265025     A=[0,1]
%o A265025     for i in [2..n]:
%o A265025         if i%2==0:
%o A265025             A.append(1-A[floor(i/2)])
%o A265025         else:
%o A265025             A.append(1)
%o A265025     return A[1:]
%o A265025 def a(n):
%o A265025     A=periodDouble(2*n-1)
%o A265025     M=matrix([[A[i] for i in [j..n+j-1]] for j in [0..n-1]])
%o A265025     return det(M)
%o A265025 [a(i) for i in [1..70]] # _Tom Edgar_, Nov 30 2015
%Y A265025 Cf. A035263, A001045.
%K A265025 sign,look
%O A265025 1,4
%A A265025 _Jeffrey Shallit_, Nov 30 2015

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