A083948 -id:A083948 - cp's OEIS frontend

A084208 G.f. A(x) defined by: A(x)^8 consists entirely of integer coefficients between 1 and 8 (A083948); A(x) is the unique power series solution with A(0)=1.

1, 1, -3, 15, -82, 484, -2992, 19110, -124979, 832234, -5621028, 38402783, -264858143, 1841221687, -12886279885, 90713376563, -641815393278, 4561172770669, -32542369727538, 232992967457839

Offset: 0

Paul D. Hanna, May 20 2003

sign

Limit a(n)/a(n+1) --> r = -0.131401689761435 where A(r)=0.

Cf. A083948, A084202-A084207, A084209-A084212.

kmax = 20;
A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
coes = CoefficientList[A[x]^8 + O[x]^(kmax + 1), x];
r = {a[0] -> 1, a[1] -> 1};
coes = coes /. r;
Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 8, 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 *)

A110636 Every 8th term of A083948 where the self-convolution 8th power is congruent modulo 16 to A083948, which consists entirely of numbers 1 through 8.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v and Paul D. Hanna, Aug 30 2005

nonn

			A(x) = 1 + 7*x + 3*x^2 + x^3 + 6*x^4 + 6*x^5 + 4*x^6 + 8*x^7 +...
A(x)^8 = 1 + 56*x + 1396*x^2 + 20392*x^3 + 193458*x^4 +...
A(x)^8 (mod 16) = 1 + 8*x + 4*x^2 + 8*x^3 + 2*x^4 + 8*x^5 +...
G(x) = 1 + 8*x + 4*x^2 + 8*x^3 + 2*x^4 + 8*x^5 + 4*x^6 +...
where G(x) is the g.f. of A083948.

Cf. A083948, A110632, A110633.

{a(n)=local(d=8,m=8,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)}

A110637 Every 4th term of A083948 where the self-convolution 4th power is congruent modulo 16 to A083948, which consists entirely of numbers 1 through 8.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v and Paul D. Hanna, Aug 30 2005

nonn

			A(x) = 1 + 2*x + 7*x^2 + 4*x^3 + 3*x^4 + 2*x^5 + x^6 +...
A(x)^4 = 1 + 8*x + 52*x^2 + 216*x^3 + 754*x^4 + 2008*x^5 +...
A(x)^4 (mod 16) = 1 + 8*x + 4*x^2 + 8*x^3 + 2*x^4 + 8*x^5 +...
G(x) = 1 + 8*x + 4*x^2 + 8*x^3 + 2*x^4 + 8*x^5 + 4*x^6 +...
where G(x) is the g.f. of A083948.

Cf. A083948, A110636, A110638.

{a(n)=local(d=4,m=8,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)}

A110638 Every 2nd term of A083948 where the self-convolution 2nd power is congruent modulo 16 to A083948, which consists entirely of numbers 1 through 8.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v and Paul D. Hanna, Aug 30 2005

nonn

			A(x) = 1 + 4*x + 2*x^2 + 4*x^3 + 7*x^4 + 8*x^5 + 4*x^6 +...
A(x)^2 = 1 + 8*x + 20*x^2 + 24*x^3 + 50*x^4 + 88*x^5 +...
A(x)^2 (mod 16) = 1 + 8*x + 4*x^2 + 8*x^3 + 2*x^4 + 8*x^5 +...
G(x) = 1 + 8*x + 4*x^2 + 8*x^3 + 2*x^4 + 8*x^5 + 4*x^6 +...
where G(x) is the g.f. of A083948.

Cf. A083948, A110636, A110637.

{a(n)=local(d=2,m=8,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)}

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

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

A083947 Integer coefficients of A(x), where 1<=a(n)<=7, such that A(x)^(1/7) consists entirely of integer coefficients.

Original entry on oeis.org

1, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 5, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 4, 7, 7, 7, 7, 7, 7, 2, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 3, 7, 7, 7, 7, 7, 7, 5, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 2, 7, 7, 7, 7, 7, 7, 5, 7, 7, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7

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, A083948, 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)^(1/7), {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

A084067 Integer coefficients of A(x), where 1<=a(n)<=12, such that A(x)^(1/12) consists entirely of integer coefficients.

Original entry on oeis.org

1, 12, 6, 4, 9, 12, 4, 12, 12, 8, 6, 12, 6, 12, 12, 12, 12, 12, 8, 12, 9, 12, 12, 12, 12, 12, 6, 12, 6, 12, 10, 12, 6, 12, 12, 12, 2, 12, 6, 8, 6, 12, 12, 12, 12, 4, 12, 12, 8, 12, 12, 8, 3, 12, 4, 12, 12, 4, 12, 12, 9, 12, 6, 4, 6, 12, 4, 12, 12, 12, 12, 12, 2, 12, 6, 12, 3, 12, 6, 12, 3, 8

Offset: 0

Paul D. Hanna, May 10 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>0. Are these sequences ever periodic?

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

Cf. A083952, A083953, A083954, A083955, A083956, A083947 A083948, A083949, A083950, A084066.

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)^(1/12), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 81}] (* Robert G. Wilson v *)

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

A084066 Least integer coefficients of A(x), where 1<=a(n)<=11, such that A(x)^(1/11) consists entirely of integer coefficients.

Original entry on oeis.org

1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 4, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 5, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 5, 11, 11, 11, 11, 11

Offset: 0

Paul D. Hanna, May 10 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>0. Are these sequences ever periodic?

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

Cf. A083952, A083953, A083954, A083955, A083956, A083947, A083948, A083949, A083950, A084067.

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)^(1/11), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 71}] (* Robert G. Wilson v *)

a(k)=0 (mod 11) when k not= 0 (mod 11); a(0)=1, a(11)=1, a(22)=7, a(33)=4, a(44)=9, a(55)=5, a(66)=5, ...

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

cp's OEIS Frontend

A084208 G.f. A(x) defined by: A(x)^8 consists entirely of integer coefficients between 1 and 8 (A083948); A(x) is the unique power series solution with A(0)=1.

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A110636 Every 8th term of A083948 where the self-convolution 8th power is congruent modulo 16 to A083948, which consists entirely of numbers 1 through 8.

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A110637 Every 4th term of A083948 where the self-convolution 4th power is congruent modulo 16 to A083948, which consists entirely of numbers 1 through 8.

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A110638 Every 2nd term of A083948 where the self-convolution 2nd power is congruent modulo 16 to A083948, which consists entirely of numbers 1 through 8.

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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.

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Formula

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

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

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A083947 Integer coefficients of A(x), where 1<=a(n)<=7, such that A(x)^(1/7) consists entirely of integer coefficients.

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A084067 Integer coefficients of A(x), where 1<=a(n)<=12, such that A(x)^(1/12) consists entirely of integer coefficients.

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