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 21 results. Next

A194519 Second coordinate of (3,7)-Lagrange pair for n.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 28 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n).. -2..3..1.-1..4..2..0..5..3..1..-1...4...2
y(n)... 1.-1..0..1.-1..0..1.-1..0..1...2...0...1

Links

Crossrefs

Cf. A194508, A194518.

Programs

  • Mathematica
    c = 3; d = 7;
x1 = {-2, 3, 1, -1, 4, 2, 0, 5, 3, 1}; y1 = {1, -1, 0, 1, -1, 0,
  1, -1, 0, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}]  (* A194518 *)
Table[y[n], {n, 1, 100}]  (* A194519 *)
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-10) - a(n-11) for n > 11.
G.f.: x*(x^9 + x^8 - 2*x^7 + x^6 + x^5 - 2*x^4 + x^3 + x^2 - 2*x + 1)/(x^11 - x^10 - x + 1). (End)
a(n) = n - 3*floor((3*n+6)/10). - Ridouane Oudra, Sep 06 2020

A194510 First coordinate of (2,5)-Lagrange pair for n.

Original entry on oeis.org

3, 1, -1, 2, 0, 3, 1, 4, 2, 0, 3, 1, 4, 2, 5, 3, 1, 4, 2, 5, 3, 6, 4, 2, 5, 3, 6, 4, 7, 5, 3, 6, 4, 7, 5, 8, 6, 4, 7, 5, 8, 6, 9, 7, 5, 8, 6, 9, 7, 10, 8, 6, 9, 7, 10, 8, 11, 9, 7, 10, 8, 11, 9, 12, 10, 8, 11, 9, 12, 10, 13, 11, 9, 12, 10, 13, 11, 14, 12, 10, 13, 11, 14, 12, 15, 13, 11, 14
Offset: 1

Views

Author

Clark Kimberling, Aug 27 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n)... 3..1.-1..2..0..3..1..4..2..0...3...1...4
y(n).. -1..0..1..0..1..0..1..0..1..2...1...2...1

Links

Crossrefs

Cf. A194508, A194511.

Programs

  • Mathematica
    c = 2; d = 5;
x1 = {3, 1, -1, 2, 0, 3, 1}; y1 = {-1, 0, 1, 0, 1, 0, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}] (* A194510 *)
Table[y[n], {n, 1, 100}] (* A194511 *)
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
G.f.: x*(-2*x^6 + 3*x^5 - 2*x^4 + 3*x^3 - 2*x^2 - 2*x + 3)/(x^8 - x^7 - x + 1). (End)
a(n) = 3*n - 5*floor((4*n + 2)/7). - Ridouane Oudra, Dec 25 2020

A194511 Second coordinate of (2,5)-Lagrange pair for n.

Original entry on oeis.org

-1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 9, 10, 11, 10, 11, 10, 11, 10, 11, 12, 11, 12, 11, 12, 11, 12, 13, 12, 13
Offset: 1

Views

Author

Clark Kimberling, Aug 27 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n)... 3..1.-1..2..0..3..1..4..2..0...3...1...4
y(n).. -1..0..1..0..1..0..1..0..1..2...1...2...1

Links

Crossrefs

Cf. A194508, A194510.

Programs

  • Mathematica
    c = 2; d = 5;
x1 = {3, 1, -1, 2, 0, 3, 1}; y1 = {-1, 0, 1, 0, 1, 0, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}] (* A194510 *)
Table[y[n], {n, 1, 100}] (* A194511 *)
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
G.f.: x*(x^6 - x^5 + x^4 - x^3 + x^2 + x - 1)/(x^8 - x^7 - x + 1). (End)
a(n) = n - 2*floor((3*n + 4)/7). - Ridouane Oudra, Dec 25 2020
G.f.: x*(1-x^2+x^3)*(x^3+x-1)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6)). - R. J. Mathar, Feb 04 2022

A194512 First coordinate of (2,7)-Lagrange pair for n.

Original entry on oeis.org

4, 1, 5, 2, -1, 3, 0, 4, 1, 5, 2, 6, 3, 0, 4, 1, 5, 2, 6, 3, 7, 4, 1, 5, 2, 6, 3, 7, 4, 8, 5, 2, 6, 3, 7, 4, 8, 5, 9, 6, 3, 7, 4, 8, 5, 9, 6, 10, 7, 4, 8, 5, 9, 6, 10, 7, 11, 8, 5, 9, 6, 10, 7, 11, 8, 12, 9, 6, 10, 7, 11, 8, 12, 9, 13, 10, 7, 11, 8, 12, 9, 13, 10, 14, 11, 8, 12, 9, 13, 10, 14, 11
Offset: 1

Views

Author

Clark Kimberling, Aug 28 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n)... 4..1..5..2.-1..3..0..4..1..5...2...6...3
y(n).. -1..0.-1..0..1..0..1..0..1..0...1...0...1

Links

Crossrefs

Cf. A194508, A194513.

Programs

  • Mathematica
    c = 2; d = 7;
x1 = {4, 1, 5, 2, -1, 3, 0, 4, 1}; y1 = {-1, 0, -1, 0, 1, 0, 1, 0, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}] (* A194512 *)
Table[y[n], {n, 1, 100}] (* A194513 *)
Table[y[n], {n, 1, 100}]
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]
LinearRecurrence[{1,0,0,0,0,0,0,0,1,-1},{4,1,5,2,-1,3,0,4,1,5},100] (* Harvey P. Dale, Dec 27 2023 *)

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-9) - a(n-10) for n > 10.
G.f.: x*(-3*x^8 + 4*x^7 - 3*x^6 + 4*x^5 - 3*x^4 - 3*x^3 + 4*x^2 - 3*x + 4)/(x^10 - x^9 - x + 1). (End)
a(n) = 4*n - 7*floor((5*n + 2)/9). - Ridouane Oudra, Dec 28 2020

A194513 Second coordinate of (2,7)-Lagrange pair for n.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 28 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n)... 4..1..5..2.-1..3..0..4..1..5...2...6...3
y(n).. -1..0.-1..0..1..0..1..0..1..0...1...0...1

Links

Crossrefs

Cf. A194508, A194512.

Programs

  • Mathematica
    c = 2; d = 7;
x1 = {4, 1, 5, 2, -1, 3, 0, 4, 1}; y1 = {-1, 0, -1, 0, 1, 0, 1, 0, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}] (* A194512 *)
Table[y[n], {n, 1, 100}] (* A194513 *)
Table[y[n], {n, 1, 100}]
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-9) - a(n-10) for n > 10.
G.f.: x*(x^8 - x^7 + x^6 - x^5 + x^4 + x^3 - x^2 + x - 1)/(x^10 - x^9 - x + 1). (End)
a(n) = n - 2*floor((4*n + 6)/9). - Ridouane Oudra, Dec 28 2020

A194514 First coordinate of (3,4)-Lagrange pair for n.

Original entry on oeis.org

-1, 2, 1, 0, -1, 2, 1, 0, 3, 2, 1, 0, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 5, 4, 3, 2, 5, 4, 3, 6, 5, 4, 3, 6, 5, 4, 7, 6, 5, 4, 7, 6, 5, 8, 7, 6, 5, 8, 7, 6, 9, 8, 7, 6, 9, 8, 7, 10, 9, 8, 7, 10, 9, 8, 11, 10, 9, 8, 11, 10, 9, 12, 11, 10, 9, 12, 11, 10, 13, 12, 11, 10, 13, 12, 11, 14, 13, 12, 11, 14, 13
Offset: 1

Views

Author

Clark Kimberling, Aug 28 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n)...-1..2..1..0.-1..2..1..0..3..2...1...0...3
y(n)... 1.-1..0..1..2..0..1..2..0..1...2...3...1

Links

Crossrefs

Cf. A194508, A194515.

Programs

  • Mathematica
    c = 3; d = 4;
x1 = {-1, 2, 1, 0, -1, 2, 1}; y1 = {1, -1, 0, 1, 2, 0, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}]  (* A194514 *)
Table[y[n], {n, 1, 100}]  (* A194515 *)
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
G.f.: x*(-x^6 + 3*x^5 - x^4 - x^3 - x^2 + 3*x - 1)/(x^8 - x^7 - x + 1). (End)
a(n) = 3*n - 4*floor((5*n + 3)/7). - Ridouane Oudra, Dec 28 2020

A194515 Second coordinate of (3,4)-Lagrange pair for n.

Original entry on oeis.org

1, -1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 6, 7, 8, 9, 7, 8, 9, 7, 8, 9, 10, 8, 9, 10, 8, 9, 10, 11, 9, 10, 11, 9, 10, 11, 12, 10, 11, 12, 10, 11, 12, 13, 11, 12, 13, 11, 12, 13, 14
Offset: 1

Views

Author

Clark Kimberling, Aug 28 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n)...-1..2..1..0.-1..2..1..0..3..2...1...0...3
y(n)... 1.-1..0..1..2..0..1..2..0..1...2...3...1

Links

Crossrefs

Cf. A194508, A194514.

Programs

  • Mathematica
    c = 3; d = 4;
x1 = {-1, 2, 1, 0, -1, 2, 1}; y1 = {1, -1, 0, 1, 2, 0, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}]  (* A194514 *)
Table[y[n], {n, 1, 100}]  (* A194515 *)
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
G.f.: x*(x^6 - 2*x^5 + x^4 + x^3 + x^2 - 2*x + 1)/(x^8 - x^7 - x + 1). (End)
a(n) = n - 3*floor((2*n + 3)/7). - Ridouane Oudra, Dec 29 2020

A194516 First coordinate of (3,5)-Lagrange pair for n.

Original entry on oeis.org

2, -1, 1, 3, 0, 2, -1, 1, 3, 0, 2, 4, 1, 3, 0, 2, 4, 1, 3, 5, 2, 4, 1, 3, 5, 2, 4, 6, 3, 5, 2, 4, 6, 3, 5, 7, 4, 6, 3, 5, 7, 4, 6, 8, 5, 7, 4, 6, 8, 5, 7, 9, 6, 8, 5, 7, 9, 6, 8, 10, 7, 9, 6, 8, 10, 7, 9, 11, 8, 10, 7, 9, 11, 8, 10, 12, 9, 11, 8, 10, 12, 9, 11, 13, 10, 12, 9, 11, 13, 10, 12, 14, 11, 13, 10
Offset: 1

Views

Author

Clark Kimberling, Aug 28 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n)... 2.-1..1..3..0..2.-1..1..3..0...2...4...1
y(n).. -1..1..0.-1..1..0..2..1..0..2...1...0...2

Links

Crossrefs

Cf. A194508, A194517.

Programs

  • Mathematica
    c = 3; d = 5;
x1 = {2, -1, 1, 3, 0, 2, -1, 1}; y1 = {-1, 1, 0, -1, 1, 0, 2, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}]  (* A194516 *)
Table[y[n], {n, 1, 100}]  (* A194517 *)
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]
LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{2,-1,1,3,0,2,-1,1,3},100] (* Harvey P. Dale, Nov 29 2024 *)

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.
G.f.: x*(2*x^7 - 3*x^6 + 2*x^5 - 3*x^4 + 2*x^3 + 2*x^2 - 3*x + 2)/(x^9 - x^8 - x + 1). (End)
a(n) = 2*n - 5*floor((3*n + 3)/8). - Ridouane Oudra, Dec 29 2020

A194517 Second coordinate of (3,5)-Lagrange pair for n.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 28 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n)... 2.-1..1..3..0..2.-1..1..3..0...2...4...1
y(n).. -1..1..0.-1..1..0..2..1..0..2...1...0...2

Links

Crossrefs

Cf. A194508, A194517.

Programs

  • Mathematica
    c = 3; d = 5;
x1 = {2, -1, 1, 3, 0, 2, -1, 1}; y1 = {-1, 1, 0, -1, 1, 0, 2, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}]  (* A194516 *)
Table[y[n], {n, 1, 100}]  (* A194517 *)
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.
G.f.: -x*(x^3 + x - 1)*(x^4 - 2*x^3 + x - 1)/(x^9 - x^8 - x + 1). (End)
a(n) = 2*n - 3*floor((5*n + 4)/8). - Ridouane Oudra, Dec 29 2020

A194518 First coordinate of (3,7)-Lagrange pair for n.

Original entry on oeis.org

-2, 3, 1, -1, 4, 2, 0, 5, 3, 1, -1, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 5, 3, 1, 6, 4, 2, 7, 5, 3, 1, 6, 4, 2, 7, 5, 3, 8, 6, 4, 2, 7, 5, 3, 8, 6, 4, 9, 7, 5, 3, 8, 6, 4, 9, 7, 5, 10, 8, 6, 4, 9, 7, 5, 10, 8, 6, 11, 9, 7, 5, 10, 8, 6, 11, 9, 7, 12, 10, 8, 6, 11, 9, 7, 12, 10, 8, 13, 11, 9, 7, 12, 10
Offset: 1

Views

Author

Clark Kimberling, Aug 28 2011

Keywords

Comments

See A194508.

Examples

			This table shows (x(n),y(n)) for 1<=n<=13:
n...... 1..2..3..4..5..6..7..8..9..10..11..12..13
x(n).. -2..3..1.-1..4..2..0..5..3..1..-1...4...2
y(n)... 1.-1..0..1.-1..0..1.-1..0..1...2...0...1

Links

Crossrefs

Cf. A194508, A194519.

Programs

  • Mathematica
    c = 3; d = 7;
x1 = {-2, 3, 1, -1, 4, 2, 0, 5, 3, 1}; y1 = {1, -1, 0, 1, -1, 0,
  1, -1, 0, 1};
x[n_] := If[n <= c + d, x1[[n]], x[n - c - d] + 1]
y[n_] := If[n <= c + d, y1[[n]], y[n - c - d] + 1]
Table[x[n], {n, 1, 100}]  (* A194518 *)
Table[y[n], {n, 1, 100}]  (* A194519 *)
r[1, n_] := n; r[2, n_] := x[n]; r[3, n_] := y[n]
TableForm[Table[r[m, n], {m, 1, 3}, {n, 1, 30}]]
LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{-2,3,1,-1,4,2,0,5,3,1,-1},100] (* Harvey P. Dale, Sep 02 2023 *)

Formula

From Chai Wah Wu, Jan 21 2020: (Start)
a(n) = a(n-1) + a(n-10) - a(n-11) for n > 11.
G.f.: x*(-2*x^9 - 2*x^8 + 5*x^7 - 2*x^6 - 2*x^5 + 5*x^4 - 2*x^3 - 2*x^2 + 5*x - 2)/(x^11 - x^10 - x + 1). (End)
a(n) = 5*n - 7*floor((7*n+3)/10). - Ridouane Oudra, Sep 06 2020
Showing 1-10 of 21 results. Next

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