cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A084212 G.f. A(x) defined by: A(x)^12 consists entirely of integer coefficients between 1 and 12 (A084067); A(x) is the unique power series solution with A(0)=1.

Original entry on oeis.org

1, 1, -5, 37, -310, 2795, -26352, 256257, -2548875, 25793149, -264579518, 2743935678, -28716005918, 302812817148, -3213908529802, 34301475340630, -367873673112308, 3962187547336323
Offset: 0

Views

Author

Paul D. Hanna, May 20 2003

Keywords

Comments

Limit a(n)/a(n+1) --> r = -0.086930814654238 where A(r)=0.

Crossrefs

Cf. A084067, A084202-A084211.

A110649 Every 2nd term of A084067 where the self-convolution 2nd power is congruent modulo 8 to A084067, which consists entirely of numbers 1 through 12.

Original entry on oeis.org

1, 6, 9, 4, 12, 6, 6, 12, 12, 8, 9, 12, 12, 6, 6, 10, 6, 12, 2, 6, 6, 12, 12, 12, 8, 12, 3, 4, 12, 12, 9, 6, 6, 4, 12, 12, 2, 6, 3, 6, 3, 6, 7, 6, 9, 8, 9, 12, 12, 12, 3, 12, 3, 6, 2, 6, 12, 2, 6, 6, 3, 12, 9, 4, 3, 12, 4, 12, 6, 2, 3, 12, 9, 6, 6, 6, 3, 6, 10, 6, 6, 6, 9, 6, 12, 12, 9, 2, 12, 6, 9
Offset: 0

Views

Author

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

Keywords

Examples

			A(x) = 1 + 6*x + 9*x^2 + 4*x^3 + 12*x^4 + 6*x^5 +...
A(x)^2 = 1 + 12*x + 54*x^2 + 116*x^3 + 153*x^4 + 228*x^5 +...
A(x)^2 (mod 8) = 1 + 4*x + 6*x^2 + 4*x^3 + x^4 + 4*x^5 +...
G(x) = 1 + 12*x + 6*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 4*x^6 +...
where G(x) is the g.f. of A084067.

Crossrefs

Cf. A084067, A110645, A110646, A110647, A110648.

Programs

  • PARI
    {a(n)=local(d=2,m=12,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)}

A110645 Every 12th term of A084067.

Original entry on oeis.org

1, 6, 12, 2, 8, 9, 2, 7, 12, 2, 3, 4, 9, 10, 12, 9, 2, 5, 12, 9, 8, 10, 1, 9, 7, 2, 9, 4, 11, 1, 10, 5, 4, 1, 12, 2, 9, 4, 5, 11, 12, 12, 9, 5, 12, 5, 3, 12, 1, 4, 6, 2, 10, 11, 5, 1, 6, 10, 12, 9, 11, 9, 9, 5, 3, 3, 7, 2, 6, 7, 5, 9, 5, 10, 12, 5, 5, 4, 4, 2, 1, 8, 7, 11, 7, 6, 1, 2, 5, 10, 9, 8, 9, 9
Offset: 0

Views

Author

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

Keywords

Crossrefs

Cf. A084067, A110646, A110647, A110648, A110649.

Programs

  • PARI
    {a(n)=local(d=12,m=12,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)}

A110646 Every 6th term of A084067.

Original entry on oeis.org

1, 4, 6, 8, 12, 10, 2, 12, 8, 4, 9, 4, 2, 6, 7, 8, 12, 12, 2, 2, 3, 4, 4, 2, 9, 6, 10, 6, 12, 2, 9, 10, 2, 2, 5, 6, 12, 2, 9, 8, 8, 8, 10, 8, 1, 4, 9, 10, 7, 10, 2, 6, 9, 10, 4, 8, 11, 2, 1, 4, 10, 2, 5, 2, 4, 2, 1, 8, 12, 2, 2, 12, 9, 2, 4, 4, 5, 8, 11, 4, 12, 4, 12, 6, 9, 12, 5, 6, 12, 10, 5, 12, 3
Offset: 0

Views

Author

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

Keywords

Crossrefs

Cf. A084067, A110645, A110647, A110648, A110649.

Programs

  • PARI
    {a(n)=local(d=6,m=12,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)}

A110647 Every 4th term of A084067 where the self-convolution 4th power is congruent modulo 8 to A084067, which consists entirely of numbers 1 through 12.

Original entry on oeis.org

1, 9, 12, 6, 12, 9, 12, 6, 6, 2, 6, 12, 8, 3, 12, 9, 6, 12, 2, 3, 3, 7, 9, 9, 12, 3, 3, 2, 12, 6, 3, 9, 3, 4, 6, 3, 9, 6, 3, 10, 6, 9, 12, 9, 12, 9, 9, 6, 2, 9, 12, 5, 3, 6, 12, 9, 6, 9, 12, 6, 8, 6, 12, 10, 9, 12, 1, 9, 3, 9, 12, 6, 7, 12, 12, 2, 9, 3, 9, 12, 12, 4, 9, 9, 11, 6, 6, 1, 9, 6, 10, 3, 12
Offset: 0

Views

Author

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

Keywords

Examples

			A(x) = 1 + 9*x + 12*x^2 + 6*x^3 + 12*x^4 + 9*x^5 +...
A(x)^4 = 1 + 36*x + 534*x^2 + 4236*x^3 + 19785*x^4 +...
A(x)^4 (mod 8) = 1 + 4*x + 6*x^2 + 4*x^3 + x^4 + 4*x^5 +...
G(x) = 1 + 12*x + 6*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 4*x^6 +...
where G(x) is the g.f. of A084067.

Crossrefs

Cf. A084067, A110645, A110646, A110648, A110649.

Programs

  • PARI
    {a(n)=local(d=4,m=12,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)}

A110648 Every third term of A084067 where the self-convolution third power is congruent modulo 9 to A084067, which consists entirely of numbers 1 through 12.

Original entry on oeis.org

1, 4, 4, 8, 6, 12, 8, 12, 12, 12, 10, 12, 2, 8, 12, 4, 8, 8, 4, 4, 9, 4, 4, 12, 2, 12, 6, 8, 7, 4, 8, 12, 12, 8, 12, 8, 2, 8, 2, 8, 3, 12, 4, 12, 4, 12, 2, 12, 9, 12, 6, 12, 10, 8, 6, 12, 12, 12, 2, 8, 9, 12, 10, 12, 2, 8, 2, 4, 5, 4, 6, 12, 12, 8, 2, 12, 9, 4, 8, 4, 8, 12, 8, 4, 10, 8, 8, 12, 1, 12
Offset: 0

Views

Author

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

Keywords

Examples

			A(x) = 1 + 4*x + 4*x^2 + 8*x^3 + 6*x^4 + 12*x^5 +...
A(x)^3 = 1 + 12*x + 60*x^2 + 184*x^3 + 450*x^4 + 948*x^5 +...
A(x)^3 (mod 9) = 1 + 3*x + 6*x^2 + 4*x^3 + 3*x^5 + 4*x^6 +...
G(x) = 1 + 12*x + 6*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 4*x^6 +...
where G(x) is the g.f. of A084067.

Crossrefs

Cf. A084067, A110645, A110646, A110647, A110649.

Programs

  • PARI
    {a(n)=local(d=3,m=12,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

Views

Author

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

Keywords

Examples

			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

Links

Crossrefs

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.

Programs

  • Mathematica
    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}]]
  • PARI
    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

Formula

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

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

Views

Author

Paul D. Hanna, May 10 2003

Keywords

Comments

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?

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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, ...

Extensions

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

A111603 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 upper right to lower left.

Original entry on oeis.org

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

Views

Author

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

Keywords

Examples

			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

Crossrefs

Cf. A111613, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
Cf. A109626, A111604.

Programs

  • Mathematica
    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}]]; g[n_, m_] := f[n][[m]]; Flatten[ Table[ f[i, n - i], {n, 15}, {i, n - 1, 1, -1}]]

A111604 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 zig-zag.

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

T(n,n)=T(n,n+2)=A111627.

Examples

			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

Crossrefs

Cf. A111613, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.
Cf. A109626, A111603.

Programs

  • Mathematica
    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}]]; g[n_, m_] := f[n][[m]];
Showing 1-10 of 11 results. Next

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