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).

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, 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, 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

Offset: 0

Author

N. J. A. Sloane and Nadia Heninger, Jul 02 2005

Keywords

• nonn
• easy
• nice

Comments

Equals A084202 read mod 2.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• 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.
• N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Crossrefs

Cf. A083952, A084202, A108335, A108337, A108340.

Programs

Maple

S:= 0: SS:= 0:
for i from 0 to 100 do
  s:= coeff(SS,x,i);
  if s = 0 or s = 3 then
     SS:= SS + 2*expand(S*x^i)+x^(2*i) mod 4; S:= S + x^i;
  fi
od:
seq(coeff(S,x,i),i=0..100); # Robert Israel, May 14 2019

Mathematica

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 *)