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A108336 Unique sequence of 1's and 0's such that (Sum_{n >= 0} a(n)*x^n)^2 mod 4 has coefficients which are all 1's and 2's (A083952).

%I A108336 #18 May 15 2019 04:49:42
%S A108336 1,1,0,1,0,1,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0,0,0,0,1,1,0,1,0,1,1,0,0,
%T A108336 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,
%U A108336 1,0,0,0,0,0,0,1,1,0,0,0,1,0,1,1,0,1,1,1,1,0,1,0,1,0,0,0,1,1,0
%N A108336 Unique sequence of 1's and 0's such that (Sum_{n >= 0} a(n)*x^n)^2 mod 4 has coefficients which are all 1's and 2's (A083952).
%C A108336 Equals A084202 read mod 2.
%H A108336 Robert Israel, <a href="/A108336/b108336.txt">Table of n, a(n) for n = 0..10000</a>
%H A108336 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006.
%H A108336 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%p A108336 S:= 0: SS:= 0:
%p A108336 for i from 0 to 100 do
%p A108336   s:= coeff(SS,x,i);
%p A108336   if s = 0 or s = 3 then
%p A108336      SS:= SS + 2*expand(S*x^i)+x^(2*i) mod 4; S:= S + x^i;
%p A108336   fi
%p A108336 od:
%p A108336 seq(coeff(S,x,i),i=0..100); # _Robert Israel_, May 14 2019
%t A108336 max = 98; (* a = A084202 *) a[n_] := a[n] = Block[{s = Sum[a[i]*x^i, {i, 0, n-1}]}, If[IntegerQ @ Last @ CoefficientList[Series[Sqrt[s + x^n], {x, 0, n}], x], 1, 2]]; Table[a[n], {n, 0, max}]; A108336 = CoefficientList[ Series[Sqrt[Sum[a[i]*x^i, {i, 0, max}]], {x, 0, max}], x] // Mod[#, 2]& (* _Jean-François Alcover_, Apr 01 2016, after _Robert G. Wilson v_ *)
%Y A108336 Cf. A083952, A084202, A108335, A108337, A108340.
%K A108336 nonn,easy,nice
%O A108336 0,1
%A A108336 _N. J. A. Sloane_ and _Nadia Heninger_, Jul 02 2005