A108336 -id:A108336 - cp's OEIS frontend

A108115 Let B(x) = Sum_{i >= 0} A108336(i)*x^i; sequence gives coefficients of B(x)^2.

1, 2, 1, 2, 2, 2, 5, 2, 2, 2, 1, 2, 1, 2, 2, 2, 6, 2, 6, 6, 2, 6, 2, 2, 2, 2, 5, 2, 6, 6, 5, 14, 5, 6, 6, 2, 5, 2, 2, 2, 2, 6, 2, 6, 6, 2, 6, 2, 2, 2, 1, 2, 1, 2, 2, 2, 5, 2, 2, 2, 1, 2, 1, 2, 2, 2, 6, 2, 6, 6, 2, 6, 2, 2, 2, 2, 6, 2, 6, 6, 6, 14, 6, 10, 6, 6, 10, 6, 14, 10, 10, 18, 10, 18, 10, 10, 14

Offset: 0

N. J. A. Sloane, Jul 03 2005

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A108336.

a(n) mod 4 = A083952(n).

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

A083952 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

1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Paul D. Hanna, May 09 2003

More generally, the sequence "integer coefficients of A(x), where 1<=a(n)<=m, such that A(x)^(1/m) consists entirely of integer coefficients", appears to have a unique solution for all m. [That is true - see Theorem 17 of Heninger-Rains-Sloane (2006). - N. J. A. Sloane, Aug 27 2015]

Is this sequence periodic? [It is not periodic for m = 2 or 3. Larger cases remain open. - N. J. A. Sloane, Aug 27 2015]

Robert G. Wilson v, 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; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A084202 (A(x)^(1/2)), A108335 (A084202 mod 4), A108336 (A084202 mod 2), A108340 (a(n) mod 2). Positions of 1's: A108783.

Cf. A083953, A083954, A083945, A083946.

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, 104}] (* Robert G. Wilson v, Nov 25 2006 *)
s = 0; a[n_] := a[n] = Block[{}, If[IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + x^n], {x, 0, n}], x], s = s + x^n; 1, s = s + 2 x^n; 2]]; Table[ a@n, {n, 0, 104}] (* Robert G. Wilson v, Sep 08 2007 *)

A083952_upto(N=99)=vector(N+1, n, if(n>1, (denominator(polcoeff(sqrt(O(x^n)+N+=x^(n-1)),n-1))>1 && N+=x^(n-1))+1, N=1)) \\ M. F. Hasler, Jan 27 2025

More terms from N. J. A. Sloane, Jul 02 2005

A108337 Positions of the odd terms in A084202, which gives the coefficients b(n) such that (B(x))^2 has coefficients 1 and 2.

Original entry on oeis.org

0, 1, 3, 5, 6, 13, 15, 16, 18, 25, 26, 28, 30, 31, 63, 65, 66, 68, 75, 76, 80, 82, 83, 85, 86, 87, 88, 90, 92, 96, 97, 99, 100, 101, 107, 108, 110, 113, 114, 115, 117, 118, 120, 121, 122, 130, 131, 132, 136, 137, 139, 141, 143, 144, 145, 147, 149, 150

Offset: 0

N. J. A. Sloane, Jul 02 2005

nonn

Also positions of 1's in A108336, which gives the coefficients of b(n) such that (B(x))^2 has coefficients 1 or 2 mod 4.

Equals A108783(n)/2.

Additional comments from Nadia Heninger, Jul 14 2007

A108335 A084202 read mod 4.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 3, 2, 2, 0, 2, 2, 0, 3, 0, 3, 1, 2, 1, 2, 2, 2, 0, 2, 0, 1, 3, 2, 1, 2, 1, 3, 2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 2, 1, 0, 3, 3, 0, 3, 0, 2, 2, 2, 0, 0, 3, 1, 2, 2, 0, 3, 0, 1, 3, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 2, 2, 1

Offset: 0

N. J. A. Sloane, Jul 02 2005

Cf. A083952, A084202, A108336.

A110627 Bisection of A083952 such that the self-convolution is congruent modulo 4 to A083952, which consists entirely of 1's and 2's.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2

Offset: 0

Robert G. Wilson v and Paul D. Hanna, Aug 08 2005

nonn

Congruent modulo 2 to A084202 and A108336; the self-convolution of A084202 equals A083952.

Cf. A083952, A111581, A084202, A108336.

{a(n)=local(p=2,A,C,X=x+x*O(x^(p*n)));if(n==0,1, A=sum(i=0,n-1,a(i)*x^(p*i))+p*x*((1-x^(p-1))/(1-X))/(1-X^p); for(k=1,p,C=polcoeff((A+k*x^(p*n))^(1/p),p*n); if(denominator(C)==1,return(k);break)))}

a(n) = A083952(2*n) for n>=0. G.f. satisfies: A(x^2) = G(x) - 2*x/(1-x^2), where G(x) is the g.f. of A083952. G.f. satisfies: A(x)^2 = A(x^2) + 2*x/(1-x^2) + 4*x^2*H(x) where H(x) is the g.f. of A111581.

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A108115 Let B(x) = Sum_{i >= 0} A108336(i)*x^i; sequence gives coefficients of B(x)^2.

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A083952 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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A108337 Positions of the odd terms in A084202, which gives the coefficients b(n) such that (B(x))^2 has coefficients 1 and 2.

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Extensions

A108335 A084202 read mod 4.

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A110627 Bisection of A083952 such that the self-convolution is congruent modulo 4 to A083952, which consists entirely of 1's and 2's.

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