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A084202 G.f. A(x) defined by: A(x)^2 consists entirely of integer coefficients between 1 and 2 (A083952); A(x) is the unique power series solution with A(0)=1.

%I A084202 #17 Jan 17 2019 17:37:01
%S A084202 1,1,0,1,0,1,-1,2,-2,4,-6,10,-16,27,-44,75,-127,218,-375,650,-1130,
%T A084202 1974,-3460,6086,-10736,18993,-33685,59882,-106683,190446,-340611,
%U A084202 610243,-1095102,1968200,-3542468,6384518,-11521308,20815942,-37651528,68176596,-123574852,224204708,-407153894
%N A084202 G.f. A(x) defined by: A(x)^2 consists entirely of integer coefficients between 1 and 2 (A083952); A(x) is the unique power series solution with A(0)=1.
%C A084202 Limit a(n)/a(n+1) -> r = -0.530852489019085 where A(r)=0.
%C A084202 Let A_n(x) be the power series formed from the first n terms of this sequence. Then a(0) = 1, a(n) = floor(1 - [x^n] (A_(n-1)(x))^2/2). Replacing 2 with a larger integer k generates the related sequences A084203-A084212. - _Charlie Neder_, Jan 16 2019
%H A084202 Paul D. Hanna, <a href="/A084202/b084202.txt">Table of n, a(n) for n = 0..1024</a>
%H A084202 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; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%t 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, 42}]; CoefficientList[ Series[ Sqrt[ Sum[ a[i]*x^i, {i, 0, 42}]], {x, 0, 42}], x] (* _Robert G. Wilson v_, Nov 11 2007 *)
%o A084202 (PARI) /* Using Charlie Neder's formula */
%o A084202 {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = floor(1 - polcoeff( Ser(A)^2, #A-1)/2) ); A[n+1]}
%o A084202 for(n=0,50, print1(a(n),", ")) \\ _Paul D. Hanna_, Jan 17 2019
%Y A084202 Cf. A083952, A084203-A084212.
%K A084202 sign
%O A084202 0,8
%A A084202 _Paul D. Hanna_, May 19 2003