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%I #23 Aug 29 2018 06:32:28
%S 0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,
%T 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,0,
%U 0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0
%N Formal inverse of the period-doubling sequence A096268.
%H Manon Stipulanti, <a href="/A317542/b317542.txt">Table of n, a(n) for n = 0..10000</a>
%H N. Rampersad and M. Stipulanti, <a href="https://arxiv.org/abs/1807.11899">The Formal Inverse of the Period-Doubling Sequence</a>, arXiv preprint arXiv:1807.11899 [math.CO], 2018.
%F a(2n) = 0 for all n >= 0; a(1) = 1, a(4n+1) = a(2n-1) for all n >= 1; a(4n+3) = a(n) for all n >= 0.
%t a[0] = 0; a[1] = 1; a[2] = 0; a[3] = 0;
%t a[n_] := If[EvenQ[n], 0,
%t If[IntegerQ[(n - 1)/4], a[2 ((n - 1)/4) - 1], a[(n - 3)/4]]]
%o (PARI) seq(n)={Vec(lift(serreverse(sum(i=1, n, (valuation(i+1, 2)%2 + O(2))*x^i) + O(x*x^n))), -(n+1))} \\ _Andrew Howroyd_, Jul 31 2018
%Y Cf. A096268, A317543, A317544.
%K nonn
%O 0
%A _Manon Stipulanti_, Jul 30 2018
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