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.

Previous Showing 11-18 of 18 results.

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

Views

Author

Paul D. Hanna, May 09 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. Are these sequences periodic?

Links

Crossrefs

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

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)^(1/7), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

Extensions

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

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, A084066.

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

Extensions

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

A083945 Least integer coefficients of A(x), where 1<=a(n)<=5, such that A(x)^(1/5) consists entirely of integer coefficients.

Original entry on oeis.org

1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 4, 5, 5, 5, 5, 3, 5, 5, 5, 5, 2, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 2, 5, 5, 5, 5, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 2, 5, 5, 5, 5, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 1, 5, 5, 5, 5, 3, 5, 5, 5, 5, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 3, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5
Offset: 0

Views

Author

Paul D. Hanna, May 09 2003

Keywords

Comments

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?

Links

Crossrefs

Cf. A083952, A083953, A083954, A083946.

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)^(1/5), {x, 0, n}], x]] != {True}, k++ ]; k]; Table[ a[n], {n, 0, 104}] (* Robert G. Wilson v *)

Extensions

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

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

A132854 Number of sequences {c(i), i=0..n} that form the initial terms of a self-convolution 4th power of an integer sequence such that 0 < c(n) <= 4*c(n-1) for n>0 with c(0)=1.

Original entry on oeis.org

1, 1, 4, 32, 736, 47600, 9901728, 6780161344, 15819971230848, 128391245362464512, 3685238521747987153664, 378871127417706380405937152, 140962622184196263047081802452992, 191428155805533938524028481989647915008
Offset: 0

Views

Author

Paul D. Hanna, Sep 19 2007, Oct 06 2007

Keywords

Comments

The minimal path in the 4-convoluted tree is A083954 and the maximal path is A132837.
Equals the number of nodes at generation n in the 4-convoluted tree, which is defined as follows: tree of all finite sequences {c(k), k=0..n} that form the initial terms of a self-convolution 4th power of some integer sequence such that 0 < c(n) <= 4*c(n-1) for n>0 with a(0)=1. - Paul D. Hanna, Oct 06 2007

Examples

			a(n) counts the nodes in generation n of the following tree.
Generations 0..3 of the 4-convoluted tree are as follows;
The path from the root is shown, with child nodes enclosed in [].
GEN.0: [1];
GEN.1: 1->[4];
GEN.2: 1-4->[2,6,10,14];
GEN.3:
1-4-2->[4,8]
1-4-6->[4,8,12,16,20,24]
1-4-10->[4,8,12,16,20,24,28,32,36,40]
1-4-14->[4,8,12,16,20,24,28,32,36,40,44,48,52,56].
Each path in the tree from the root node forms the initial terms of a self-convolution 4th power of a sequence of integer terms.

Links

Crossrefs

Cf. A132852, A132853, A132855, A132856; A083954, A132837.

Extensions

Extended by Martin Fuller, Sep 24 2007

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]];

A112570 G.f. A(x) satisfies: A(x)^2 equals the g.f. of A110630, which consists entirely of numbers 1 through 4.

Original entry on oeis.org

1, 1, 1, 1, -1, 2, 0, 1, -2, 5, -5, 4, -6, 18, -30, 35, -43, 84, -167, 261, -352, 545, -1010, 1790, -2783, 4207, -7025, 12464, -21071, 33567, -54154, 92317, -159366, 266150, -435285, 725260, -1239404, 2112351, -3535532, 5894852, -9964767, 17008752, -28880694, 48645873
Offset: 0

Views

Author

Paul D. Hanna, Sep 14 2005

Keywords

Comments

A110630 is formed from every 2nd term of A083954, which also consists entirely of numbers 1 through 4.

Examples

			A(x) = 1 + x + x^2 + x^3 - x^4 + 2*x^5 + x^7 - 2*x^8 + 5*x^9 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + x^4 + 4*x^5 + 3*x^6 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 27*x^4 + 36*x^5 +...
A(x)^4 (mod 8) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 +...
G(x) = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 4*x^5 + 4*x^6 +...
where G(x) is the g.f. of A083954.

Crossrefs

Cf. A110630, A083954.

Programs

  • PARI
    {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(Ser(vector(n+1,i,polcoeff(A,d*(i-1))))^(1/2),n)}

Formula

G.f. A(x) satisfies: A(x)^4 (mod 8) = g.f. of A083954.
Previous Showing 11-18 of 18 results.

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