A108337 -id:A108337 - cp's OEIS frontend

A108338 First difference of A108337.

1, 2, 2, 1, 7, 2, 1, 2, 7, 1, 2, 2, 1, 32, 2, 1, 2, 7, 1, 4, 2, 1, 2, 1, 1, 1, 2, 2, 4, 1, 2, 1, 1, 6, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 8, 1, 1, 4, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 2, 1, 3, 1, 1, 4, 1, 6, 1, 2, 3, 3, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 3, 3, 2, 3, 1, 3, 1, 1, 2, 1

Offset: 0

N. J. A. Sloane, Jul 02 2005

nonn

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).

Original entry on oeis.org

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

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

Equals A084202 read mod 2.

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.

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

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

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 *)

A108783 Positions of 1's in A083952, where A083952 gives the integer coefficients a(n) of A(x), where a(n) = 1 or 2 for all n, such that A(x)^(1/2) has only integer coefficients.

Original entry on oeis.org

0, 2, 6, 10, 12, 26, 30, 32, 36, 50, 52, 56, 60, 62, 126, 130, 132, 136, 150, 152, 160, 164, 166, 170, 172, 174, 176, 180, 184, 192, 194, 198, 200, 202, 214, 216, 220, 226, 228, 230, 234, 236, 240, 242, 244, 260, 262, 264, 272, 274, 278, 282, 286

Offset: 1

N. J. A. Sloane, following a suggestion from Paul D. Hanna, Jun 30 2005

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..1356

Cf. A083952, A108337, A108338. See A111363 for another version.

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]]; Union@ Table[ If[ a[n] == 1, n, 0], {n, 0, 300}] (* Robert G. Wilson v, Nov 25 2006 *)

A108783_upto(N=200)=[k-1 | k<-select(t->t==1, A083952_upto(N),1)] \\ M. F. Hasler, Jan 27 2025

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A108338 First difference of A108337.

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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).

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A108783 Positions of 1's in A083952, where A083952 gives the integer coefficients a(n) of A(x), where a(n) = 1 or 2 for all n, such that A(x)^(1/2) has only integer coefficients.

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