A083954 -id:A083954 - cp's OEIS frontend

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.

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 *)

A110630 Every 2nd term of A083954 such that the self-convolution 2nd power is congruent modulo 8 to A083954, which consists entirely of numbers 1 through 4.

Original entry on oeis.org

1, 2, 3, 4, 1, 4, 3, 4, 3, 4, 2, 2, 4, 4, 3, 2, 2, 2, 3, 2, 3, 2, 4, 2, 2, 4, 2, 4, 2, 2, 1, 4, 1, 2, 4, 4, 1, 2, 3, 4, 4, 4, 3, 4, 2, 2, 2, 2, 1, 4, 1, 2, 3, 2, 4, 4, 1, 4, 1, 4, 2, 2, 3, 4, 2, 4, 2, 4, 3, 4, 4, 2, 4, 2, 1, 2, 4, 4, 4, 4, 1, 2, 4, 4, 2, 2, 3, 4, 1, 2, 2, 4, 1, 2, 4, 4, 3, 2, 3, 4, 1, 4, 4, 4, 3

Offset: 0

Robert G. Wilson v and Paul D. Hanna, Aug 09 2005

nonn

			A(x) = 1 + 2*x + 3*x^2 + 4*x^3 + x^4 + 4*x^5 + 3*x^6 + 4*x^7 +...
A(x)^2 = 1 + 4*x + 10*x^2 + 20*x^3 + 27*x^4 + 36*x^5 + 44*x^6 +...
A(x)^2 (mod 8) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 +...
G083954(x) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 + 4*x^6 +...
where G083954(x) is the g.f. of A083954.

Cf. A083954, A110629.

{a(n)=local(d=2,m=4,A=1+m*x); for(j=2,d*n, for(k=1,m,t=polcoeff((A+k*x^j+x*O(x^j))^(1/m),j); if(denominator(t)==1,A=A+k*x^j;break)));polcoeff(A,d*n)}

a(n) = A083954(2*n) for n>=0.

A110629 Every 4th term of A083954 such that the self-convolution 4th power is congruent modulo 8 to A083954, which consists entirely of numbers 1 through 4.

Original entry on oeis.org

1, 3, 1, 3, 3, 2, 4, 3, 2, 3, 3, 4, 2, 2, 2, 1, 1, 4, 1, 3, 4, 3, 2, 2, 1, 1, 3, 4, 1, 1, 2, 3, 2, 2, 3, 4, 4, 1, 4, 4, 1, 4, 2, 3, 1, 2, 1, 4, 3, 3, 1, 4, 3, 3, 2, 3, 4, 2, 3, 4, 1, 2, 1, 3, 4, 3, 4, 1, 4, 2, 2, 3, 1, 4, 3, 2, 1, 4, 3, 4, 4, 2, 1, 4, 1, 4, 4, 2, 4, 4, 1, 3, 3, 4, 1, 1, 1, 4, 3, 2, 1, 3, 1, 2, 2

Offset: 0

Robert G. Wilson v and Paul D. Hanna, Aug 09 2005

nonn

			A(x) = 1 + 3*x + x^2 + 3*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + ...
A(x)^4 = 1 + 12*x + 58*x^2 + 156*x^3 + 315*x^4 + 620*x^5 +...
A(x)^4 (mod 8) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 +...
G083954(x) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 +...
where G083954(x) is the g.f. of A083954.

Cf. A083954, A110630.

{a(n)=local(d=4,m=4,A=1+m*x); for(j=2,d*n, for(k=1,m,t=polcoeff((A+k*x^j+x*O(x^j))^(1/m),j); if(denominator(t)==1,A=A+k*x^j;break)));polcoeff(A,d*n)}

a(n) = A083954(4*n) for n>=0.

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

A109626 Consider the array T(n,m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read from lower left to upper right.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 5, 2, 1, 2, 1, 1, 6, 5, 4, 3, 2, 1, 1, 7, 3, 5, 3, 3, 1, 1, 1, 8, 7, 2, 5, 4, 3, 2, 1, 1, 9, 4, 7, 3, 1, 4, 3, 2, 1, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 11, 5, 3, 2, 7, 6, 5, 1, 3, 1, 1, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 13, 6, 11, 10, 9, 4, 1, 3, 5

Offset: 1

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

			Table begins:
\k...0...1...2...3...4...5...6...7...8...9..10..11..12..13
n\
 1|  1   1   1   1   1   1   1   1   1   1   1   1   1   1
 2|  1   2   1   2   2   2   1   2   2   2   1   2   1   2
 3|  1   3   3   1   3   3   3   3   3   3   3   3   1   3
 4|  1   4   2   4   3   4   4   4   1   4   4   4   3   4
 5|  1   5   5   5   5   1   5   5   5   5   4   5   5   5
 6|  1   6   3   2   3   6   6   6   3   4   6   6   6   6
 7|  1   7   7   7   7   7   7   1   7   7   7   7   7   7
 8|  1   8   4   8   2   8   4   8   7   8   8   8   4   8
 9|  1   9   9   3   9   9   3   9   9   1   9   9   6   9
10|  1  10   5  10  10   2   5  10  10  10   3  10   5  10
11|  1  11  11  11  11  11  11  11  11  11  11   1  11  11
12|  1  12   6   4   9  12   4  12  12   8   6  12   6  12
13|  1  13  13  13  13  13  13  13  13  13  13  13  13   1
14|  1  14   7  14   7  14  14   2   7  14  14  14  14  14
15|  1  15  15   5  15   3  10  15  15  10  15  15   5  15
16|  1  16   8  16   4  16   8  16  10  16   8  16  12  16

G. C. Greubel, Antidiagonals n = 1..100, flattened

Rows: A000012, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
Columns: A000012, A111627, A026741, A051176, A111607, A060791, A111608, A106608, A111609, A111610, A111611, A106612, A106614, A106618, A106620.
Diagonals: A000027 (main), A111614 (first upper), A111627 (2nd), A111615 (3rd), A111618 (first lower), A111623 (2nd).
Other diagonals: A005408 (T(2*n-1, n)), A111626, A111627, A111628, A111629, A111630.
Cf. A111603, A111604.

f[n_]:= f[n]= Block[{a}, a[0] = 1; a[l_]:= a[l]= Block[{k = 1, s = Sum[ a[i]*x^i, {i,0,l-1}]}, While[ IntegerQ[Last[CoefficientList[Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j,0,32}]];
T[n_, m_]:= f[n][[m]];
Flatten[Table[T[i,n-i], {n,15}, {i,n-1,1,-1}]]

A109626_row(n, len=40)={my(A=1, m); vector(len, k, if(k>m=1, while(denominator(polcoeff(sqrtn(O(x^k)+A+=x^(k-1), n), k-1))>1, m++); m, 1))} \\ M. F. Hasler, Jan 27 2025

When m is prime, column m is T(n,m) = n/gcd(m, n) = numerator of n/(n+m). - M. F. Hasler, Jan 27 2025

A083953 Least integer coefficients of A(x), where 1<=a(n)<=3, such that A(x)^(1/3) consists entirely of integer coefficients.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally, "least 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. Is this sequence periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..5000.
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, A083954, A083945, A083946.

a[0]=1; a[n_] :=a[n] = Block[{k=1, s=Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/3), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v, Jul 25 2005 *)

a(k)=1 at k=0, 3, 12, 21, 51, 57, 60, 63, 66, ...; a(k)=2 at k=15, 18, 24, 30, 39, 42, 48, 54, ...

More terms from Robert G. Wilson v, Jul 25 2005

A083948 Integer coefficients of A(x), where 1<=a(n)<=8, such that A(x)^(1/8) consists entirely of integer coefficients.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, May 09 2003

nonn

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. Are these sequences periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

Cf. A083952, A083953, A083954, A083955, A083956, A083947, A083949, A083950.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/8), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A083949 Integer coefficients of A(x), where 1<=a(n)<=9, such that A(x)^(1/9) consists entirely of integer coefficients.

Original entry on oeis.org

1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 9, 9, 6, 9, 9, 6, 9, 9, 9, 9, 9, 6, 9, 9, 6, 9, 9, 9, 9, 9, 3, 9, 9, 3, 9, 9, 2, 9, 9, 6, 9, 9, 6, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 5, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9, 9, 6, 9, 9, 6, 9, 9, 5, 9, 9, 9, 9, 9, 9, 9, 9, 3, 9, 9, 9, 9, 9, 9, 9, 9, 1, 9, 9, 6, 9, 9, 6, 9, 9, 7, 9, 9, 6, 9, 9

Offset: 0

Paul D. Hanna, May 09 2003

nonn

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. Are these sequences periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

Cf. A083952, A083953, A083954, A083955, A083956, A083947, A083948, A083950.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/9), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A083950 Integer coefficients of A(x), where 1<=a(n)<=10, such that A(x)^(1/10) consists entirely of integer coefficients.

Original entry on oeis.org

1, 10, 5, 10, 10, 2, 5, 10, 10, 10, 3, 10, 5, 10, 10, 2, 10, 10, 10, 10, 5, 10, 5, 10, 5, 8, 5, 10, 5, 10, 8, 10, 10, 10, 10, 4, 5, 10, 10, 10, 7, 10, 10, 10, 5, 2, 10, 10, 5, 10, 7, 10, 5, 10, 5, 4, 10, 10, 10, 10, 7, 10, 10, 10, 10, 2, 5, 10, 5, 10, 9, 10, 5, 10, 5, 6, 5, 10, 10, 10, 8

Offset: 0

Paul D. Hanna, May 09 2003

nonn

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. Are these sequences periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.

Cf. A083952, A083953, A083954, A083952, A083956, A083947, A083948, A083949.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/10), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 80}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

A083946 Least integer coefficients of A(x), where 1<=a(n)<=6, such that A(x)^(1/6) consists entirely of integer coefficients.

Original entry on oeis.org

1, 6, 3, 2, 3, 6, 6, 6, 3, 4, 6, 6, 6, 6, 3, 4, 6, 6, 3, 6, 6, 2, 3, 6, 6, 6, 3, 4, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 3, 4, 6, 6, 4, 6, 6, 2, 6, 6, 4, 6, 3, 2, 3, 6, 6, 6, 3, 4, 3, 6, 3, 6, 3, 4, 6, 6, 2, 6, 3, 6, 3, 6, 1, 6, 6, 4, 6, 6, 2, 6, 6, 2, 6, 6, 3, 6, 3, 4, 6, 6, 1, 6, 6, 6, 6, 6, 6, 6, 3, 2, 6, 6, 6, 6, 3

Offset: 0

Paul D. Hanna, May 09 2003

nonn

More generally, "least 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>0. Is this sequence periodic?

Robert G. Wilson v, Table of n, a(n) for n = 0..3000.
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, A083953, A083954, A083945.

a[0] = 1; a[n_] := a[n] = Block[{k = 1, s = Sum[a[i]*x^i, {i, 0, n-1}]}, While[ Union[ IntegerQ /@ CoefficientList[ Series[(s+k*x^n)^(1/6), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jul 26 2005

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A110630 Every 2nd term of A083954 such that the self-convolution 2nd power is congruent modulo 8 to A083954, which consists entirely of numbers 1 through 4.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A110629 Every 4th term of A083954 such that the self-convolution 4th power is congruent modulo 8 to A083954, which consists entirely of numbers 1 through 4.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A109626 Consider the array T(n,m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read from lower left to upper right.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A083953 Least integer coefficients of A(x), where 1<=a(n)<=3, such that A(x)^(1/3) consists entirely of integer coefficients.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A083948 Integer coefficients of A(x), where 1<=a(n)<=8, such that A(x)^(1/8) consists entirely of integer coefficients.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A083949 Integer coefficients of A(x), where 1<=a(n)<=9, such that A(x)^(1/9) consists entirely of integer coefficients.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs