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.

1, 1, 0, 1, 0, 1, -1, 2, -2, 4, -6, 10, -16, 27, -44, 75, -127, 218, -375, 650, -1130, 1974, -3460, 6086, -10736, 18993, -33685, 59882, -106683, 190446, -340611, 610243, -1095102, 1968200, -3542468, 6384518, -11521308, 20815942, -37651528, 68176596, -123574852, 224204708, -407153894

Offset: 0

Author

Paul D. Hanna, May 19 2003

Keywords

• sign

Comments

Limit a(n)/a(n+1) -> r = -0.530852489019085 where A(r)=0.

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

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1024
• N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Crossrefs

Cf. A083952, A084203-A084212.

Programs

Mathematica

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 *)

PARI

/* Using Charlie Neder's formula */
{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]}
for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Jan 17 2019