A084203 -id:A084203 - cp's OEIS frontend

A104405 A084203 (the cube root of A083953) read mod 3.

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

Offset: 0

N. J. A. Sloane and Nadia Heninger, Jul 19 2005

nonn

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.

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.

Original entry on oeis.org

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

Paul D. Hanna, May 19 2003

sign

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

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.

Cf. A083952, A084203-A084212.

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 *)

/* 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

A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -1, 3, -7, 20, -58, 177, -554, 1769, -5739, 18866, -62684, 210146, -709882, 2413743, -8253995, 28366316, -97916761, 339326189, -1180068800, 4116957243, -14404398636, 50530280752, -177684095927, 626181400993, -2211215950469, 7823025701314, -27724997048327

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) -> r = -0.269562488839799 where A(r)=0.

Robert Israel, Table of n, a(n) for n = 0..1000
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. A083954, A084202, A084203, A084205-A084212.

g:= 1: a[0]:= 1:
for n from 1 to 50 do
  a[n]:= -floor((coeff(g^4,x,n)-1)/4);
  g:= g + a[n]*x^n;
od:
seq(a[n],n=0..50); # Robert Israel, Sep 04 2019

kmax = 30;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^4 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1}; coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 4, a[k-1], Integers] // ToRules];
coes = coes /. r, {k, 3, kmax + 1}];
Table[a[k], {k, 0, kmax}] /. r (* Jean-François Alcover, Jul 26 2018 *)

A110628 Trisection of A083953 such that the self-convolution cube is congruent modulo 9 to A083953, which consists entirely of 1's, 2's and 3's.

Original entry on oeis.org

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

Offset: 0

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

nonn

Congruent modulo 3 to A084203 and A104405; the self-convolution cube of A084203 equals A083953.

Cf. A083953, A111582, A084203, A104405.

{a(n)=local(p=3,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) = A083953(3*n) for n>=0. G.f. satisfies: A(x^3) = G(x) - 3*x*(1+x)/(1-x^3), where G(x) is the g.f. of A083953. G.f. satisfies: A(x)^3 = A(x^3) + 3*x*(1+x)/(1-x^3) + 9*x^2*H(x) where H(x) is the g.f. of A111582.

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A104405 A084203 (the cube root of A083953) read mod 3.

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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.

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A084204 G.f. A(x) defined by: A(x)^4 consists entirely of integer coefficients between 1 and 4 (A083954); A(x) is the unique power series solution with A(0)=1.

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A110628 Trisection of A083953 such that the self-convolution cube is congruent modulo 9 to A083953, which consists entirely of 1's, 2's and 3's.

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Author

Keywords

Comments

Crossrefs

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