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 1-8 of 8 results.

A207191 Numbers that match even polynomials among the monic polynomials over {-1,0,1}, ordered as at A206821.

Original entry on oeis.org

1, 4, 5, 8, 26, 27, 30, 31, 42, 45, 46, 120, 121, 124, 125, 136, 137, 140, 141, 184, 187, 188, 199, 200, 203, 204, 502, 503, 506, 507, 518, 519, 522, 523, 566, 567, 570, 571, 582, 583, 586, 587, 758, 761, 762, 773, 774, 777, 778, 821, 822, 825, 826
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2012

Keywords

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
Numbers n for which y(n,-x)=y(n,x): 1,4,5,8,26,...
Numbers n for which y(n,-x)=-y(n,x): 2,11,13,20,...

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
Flatten[Position[y - (y /. x -> -x), 0]]  (* A207191 *)
Flatten[Position[y + (y /. x -> -x), 0]]  (* A207192  *)

A207192 Numbers that match odd polynomials among the monic polynomials over {-1,0,1}, ordered as at A206821.

Original entry on oeis.org

2, 11, 13, 20, 57, 59, 65, 67, 90, 96, 98, 247, 249, 255, 257, 279, 281, 287, 289, 376, 382, 384, 406, 408, 414, 416, 1013, 1015, 1021, 1023, 1045, 1047, 1053, 1055, 1141, 1143, 1149, 1151, 1173, 1175, 1181, 1183, 1526, 1532, 1534, 1556, 1558
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2012

Keywords

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
Numbers n for which y(n,-x)=y(n,x): 1,4,5,8,26,...
Numbers n for which y(n,-x)=-y(n,x): 2,11,13,20,...

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
Flatten[Position[y - (y /. x -> -x), 0]]  (* A207191 *)
Flatten[Position[y + (y /. x -> -x), 0]]  (* A207192  *)

A206829 Number of distinct irreducible factors of the polynomial y(n,x) defined at A206821.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 12 2012

Keywords

Comments

The first 6 polynomials: 1, x, 1+x, x^2, x^2-1, x^2-x, representing an ordering of the monic polynomials having coefficients in {-1,0,1}; see A206821.

Examples

			y(5,x) = (x-1)(x+1), so a(5)=2.

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1, Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 8}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[-1 + Length[FactorList[y[[n]]]],
{n, 1, 120}]  (* A206829 *)

A206822 Numbers that match non-irreducible monic polynomials having coefficients in {-1,0,1}; complement of A206821.

Original entry on oeis.org

1, 4, 5, 6, 9, 11, 12, 13, 15, 17, 19, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 37, 38, 39, 40, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 89, 90, 91, 92, 94, 95, 96, 98, 100, 101, 102
Offset: 1

Views

Author

Clark Kimberling, Feb 12 2012

Keywords

Comments

See A206821.

Examples

			(See A206821.)

Crossrefs

Cf. A206821, A205783, A206331.

Programs

A207187 Numbers matching polynomials y(k,x) that have x+1 as a factor; see Comments.

Original entry on oeis.org

3, 5, 9, 13, 19, 22, 25, 27, 30, 33, 39, 43, 49, 52, 55, 59, 65, 68, 71, 83, 89, 92, 95, 101, 104, 107, 110, 113, 116, 119, 121, 124, 127, 133, 136, 139, 142, 145, 148, 151, 157, 169, 172, 175, 181, 185, 191, 194, 197, 209, 215, 218, 221, 224, 227, 230
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2012

Keywords

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
The list exemplifies these sequences:
A207187 (multiples of x + 1): 3,5,9,13,...
A207188 (multiples of x): 2,4,6,9,11,13,...
A207189 (multiples of x - 1): 5,6,12,13,...
A207190 (multiples of x^2 + 1): 8,20,25,27,...

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* monic polynomials over {-1,0,1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[y[[n]] /. x -> -1, {n, 1, 300}];
Flatten[Position[%, 0]]  (* A207187 *)
Table[y[[n]] /. x -> 0, {n, 1, 300}] ;
Flatten[Position[%, 0]]  (* A207188 *)
Table[y[[n]] /. x -> 1, {n, 1, 1200}] ;
Flatten[Position[%, 0]]  (* A207189 *)
Table[y[[n]] /. x -> I, {n, 1, 600}] ;
Flatten[Position[%, 0]]  (* A207190 *)

A207188 Numbers matching polynomials y(k,x) that have x as a factor; see Comments.

Original entry on oeis.org

2, 4, 6, 9, 11, 13, 15, 17, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2012

Keywords

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
The list exemplifies these sequences:
A207187 (multiples of x + 1): 3,5,9,13,...
A207188 (multiples of x): 2,4,6,9,11,13,...
A207189 (multiples of x - 1): 5,6,12,13,...
A207190 (multiples of x^2 + 1): 8,20,25,27,...

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[y[[n]] /. x -> -1, {n, 1, 300}];
Flatten[Position[%, 0]]  (* A207187 *)
Table[y[[n]] /. x -> 0, {n, 1, 300}] ;
Flatten[Position[%, 0]]  (* A207188 *)
Table[y[[n]] /. x -> 1, {n, 1, 1200}] ;
Flatten[Position[%, 0]]  (* A207189 *)
Table[y[[n]] /. x -> I, {n, 1, 600}] ;
Flatten[Position[%, 0]]  (* A207190 *)

A207189 Numbers matching polynomials y(k,x) that have x-1 as a factor; see Comments.

Original entry on oeis.org

5, 6, 12, 13, 15, 27, 28, 30, 34, 58, 59, 61, 65, 73, 121, 122, 124, 128, 136, 152, 248, 249, 251, 255, 263, 279, 311, 503, 504, 506, 510, 518, 534, 566, 630, 1014, 1015, 1017, 1021, 1029, 1045, 1077, 1141, 1269, 2037, 2038, 2040, 2044, 2052, 2068
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2012

Keywords

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
The list exemplifies these sequences:
A207187 (multiples of x + 1): 3,5,9,13,...
A207188 (multiples of x): 2,4,6,9,11,13,...
A207189 (multiples of x - 1): 5,6,12,13,...
A207190 (multiples of x^2 + 1): 8,20,25,27,...

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[y[[n]] /. x -> -1, {n, 1, 300}];
Flatten[Position[%, 0]]  (* A207187 *)
Table[y[[n]] /. x -> 0, {n, 1, 300}] ;
Flatten[Position[%, 0]]  (* A207188 *)
Table[y[[n]] /. x -> 1, {n, 1, 1200}] ;
Flatten[Position[%, 0]]  (* A207189 *)
Table[y[[n]] /. x -> I, {n, 1, 600}] ;
Flatten[Position[%, 0]]  (* A207190 *)

A207190 Numbers matching polynomials y(k,x) that have x^2 + 1 as a factor; see Comments.

Original entry on oeis.org

8, 20, 25, 27, 37, 45, 55, 59, 64, 79, 96, 101, 116, 124, 134, 164, 184, 194, 199, 204, 209, 214, 224, 239, 244, 255, 260, 275, 320, 335, 340, 376, 381, 396, 406, 411, 416, 421, 426, 436, 441, 456, 461, 471, 481, 486, 491, 496, 501, 503, 513, 518
Offset: 1

Views

Author

Clark Kimberling, Feb 16 2012

Keywords

Comments

The polynomials y(k,x) range through all monic polynomials with coefficients in {-1,0,1}, ordered as at A206821.

Examples

			The first 13 polynomials:
1 .... 1
2 .... x
3 .... x + 1
4 .... x^2
5 .... x^2 - 1
6 .... x^2 - x
7 .... x^2 - x - 1
8 .... x^2 + 1
9 .... x^2 + x
10 ... x^2 + x + 1
11 ... x^3
12 ... x^3 - 1
13 ... x^3 - x
The list exemplifies these sequences:
A207187 (multiples of x + 1): 3,5,9,13,...
A207188 (multiples of x): 2,4,6,9,11,13,...
A207189 (multiples of x - 1): 5,6,12,13,...
A207190 (multiples of x^2 + 1): 8,20,25,27,...

Crossrefs

Cf. A206821.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 2000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]]
q2[n_] := p[n - f[k] + 2]
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,
  Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 10}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
TableForm[Table[{n, y[[n]], Factor[y[[n]]]}, {n, 1, 10}]]
Table[y[[n]] /. x -> -1, {n, 1, 300}];
Flatten[Position[%, 0]]  (* A207187 *)
Table[y[[n]] /. x -> 0, {n, 1, 300}] ;
Flatten[Position[%, 0]]  (* A207188 *)
Table[y[[n]] /. x -> 1, {n, 1, 1200}] ;
Flatten[Position[%, 0]]  (* A207189 *)
Table[y[[n]] /. x -> I, {n, 1, 600}] ;
Flatten[Position[%, 0]]  (* A207190 *)
Showing 1-8 of 8 results.

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