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 1-6 of 6 results.

A210873 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 1, 2, 8, 5, 1, 1, 2, 6, 17, 6, 1, 1, 2, 5, 18, 31, 7, 1, 1, 2, 5, 14, 47, 51, 8, 1, 1, 2, 5, 13, 41, 107, 78, 9, 1, 1, 2, 5, 13, 35, 115, 218, 113, 10, 1, 1, 2, 5, 13, 34, 98, 296, 407, 157, 11, 1, 1, 2, 5, 13, 34, 90, 276, 695, 709, 211, 12
Offset: 1

Views

Author

Clark Kimberling, Mar 29 2012

Keywords

Comments

Column 1: 1,1,1,1,1,1,1,1,1...
Row sums: A083318 (1+2^n)
Alternating row sums: A137470
Limiting row: 1,1,2,5,13,34,..., odd-indexed Fibonacci numbers
If the term in row n and column k is written as U(n,k), then U(n,n-1)=A105163.
For a discussion and guide to related arrays, see A208510.

Examples

			First six rows:
1
1...2
1...1...3
1...1...3....4
1...1...2....8...5
1...1...2....6...17...6
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2

Crossrefs

Cf. A210872, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210872 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210873 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)

Formula

For a discussion and guide to related arrays, see A208510.
u(n,x)=x*u(n-1,x)+v(n-1,x)-1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210215 Triangle of coefficients of polynomials u(n,x) jointly generated with A210216; see the Formula section.

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 2, 5, 7, 1, 2, 5, 12, 11, 1, 2, 5, 13, 26, 16, 1, 2, 5, 13, 33, 51, 22, 1, 2, 5, 13, 34, 79, 92, 29, 1, 2, 5, 13, 34, 88, 176, 155, 37, 1, 2, 5, 13, 34, 89, 221, 365, 247, 46, 1, 2, 5, 13, 34, 89, 232, 530, 709, 376, 56, 1, 2, 5, 13, 34, 89, 233, 596
Offset: 1

Views

Author

Clark Kimberling, Mar 19 2012

Keywords

Comments

Limiting row: odd-indexed Fibonacci numbers, (A122367, A001519)
n-th row sum: -1+2^n
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
2...1
2...4...1
2...5...7....1
2...5...12...11...1
First three polynomials u(n,x): 1, 2 + x, 2 + 4x + x^2.

Crossrefs

Cf. A210216, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210215 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210216 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

Formula

u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=xu(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210216 Triangle of coefficients of polynomials v(n,x) jointly generated with A210215; see the Formula section.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 3, 7, 4, 1, 3, 8, 14, 5, 1, 3, 8, 20, 25, 6, 1, 3, 8, 21, 46, 41, 7, 1, 3, 8, 21, 54, 97, 63, 8, 1, 3, 8, 21, 55, 133, 189, 92, 9, 1, 3, 8, 21, 55, 143, 309, 344, 129, 10, 1, 3, 8, 21, 55, 144, 364, 674, 591, 175, 11, 1, 3, 8, 21, 55, 144, 376, 894
Offset: 1

Views

Author

Clark Kimberling, Mar 19 2012

Keywords

Comments

Limiting row: even-indexed Fibonacci numbers, A001906.
n-th row sum: -1+2*n
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...2
1...3...3
1...3...7...4
1...3...8...14...5
First three polynomials v(n,x): 1, 1 + 2x , 1 + 3x + 3x^2.

Crossrefs

Cf. A210215, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210215 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210216 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

Formula

u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=xu(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210872 Triangle of coefficients of polynomials u(n,x) jointly generated with A210873; see the Formula section.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 1, 5, 1, 0, 1, 4, 9, 1, 0, 1, 3, 12, 14, 1, 0, 1, 3, 9, 29, 20, 1, 0, 1, 3, 8, 27, 60, 27, 1, 0, 1, 3, 8, 22, 74, 111, 35, 1, 0, 1, 3, 8, 21, 63, 181, 189, 44, 1, 0, 1, 3, 8, 21, 56, 178, 399, 302, 54, 1, 0, 1, 3, 8, 21, 55, 154, 474, 806, 459, 65, 1, 0, 1
Offset: 1

Views

Author

Clark Kimberling, Mar 29 2012

Keywords

Comments

Column 1: 1,0,0,0,0,0,0,0,0,...
Row sums: A000225 (-1+2^n)
Alternating row sums: (-1)*A077973
Limiting row: 0,1,3,8,21,..., even-indexed Fibonacci numbers
If the term in row n and column k is written as U(n,k), then U(n,n-1)=A000096 and U(n,n-2)=A086274.
For a discussion and guide to related arrays, see A208510.

Examples

			First six rows:
1
0...1
0...2...1
0...1...5...1
0...1...4...9....1
0...1...3...12...14...1
First three polynomials u(n,x): 1, x, 2x + x^2.

Crossrefs

Cf. A210873, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210872 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210873 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A083318 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* -A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A137470 *)

Formula

u(n,x)=x*u(n-1,x)+v(n-1,x)-1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also, u(n,x)=2x*u(n-1,x)+(x-x^2)*u(n-2,x)+x, where u(2,x)=x.

A210876 Triangle of coefficients of polynomials u(n,x) jointly generated with A210877; see the Formula section.

Original entry on oeis.org

1, 2, 1, 1, 5, 1, 1, 4, 9, 1, 1, 3, 12, 14, 1, 1, 3, 9, 29, 20, 1, 1, 3, 8, 27, 60, 27, 1, 1, 3, 8, 22, 74, 111, 35, 1, 1, 3, 8, 21, 63, 181, 189, 44, 1, 1, 3, 8, 21, 56, 178, 399, 302, 54, 1, 1, 3, 8, 21, 55, 154, 474, 806, 459, 65, 1, 1, 3, 8, 21, 55, 145, 430, 1169
Offset: 1

Views

Author

Clark Kimberling, Mar 30 2012

Keywords

Comments

For n>2, each row begins with 1 and ends with 1. If the term in row n and column k is denoted by U(n,k), then U(n,n-2)=A000096(n-1) and U(n,n-3)=A086274(n-1).
Row sums: A000225 (-1+2^n)
Alternating row sums: A077973
Limiting row: 1,3,8,21,55,..., even-indexed Fibonacci numbers
For a discussion and guide to related arrays, see A208510.

Examples

			First six rows:
1
2...1
1...5...1
1...4...9....1
1...3...12...14...1
1...3...9....29...20...1
First three polynomials u(n,x): 1, 2 + x, 1 + 5x + x^2.

Crossrefs

Cf. A210877, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + x;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210876 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210877 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

Formula

u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+x,
where u(1,x)=1, v(1,x)=1.

A210877 Triangle of coefficients of polynomials v(n,x) jointly generated with A210876; see the Formula section.

Original entry on oeis.org

1, 0, 3, 0, 3, 4, 0, 2, 8, 5, 0, 2, 6, 17, 6, 0, 2, 5, 18, 31, 7, 0, 2, 5, 14, 47, 51, 8, 0, 2, 5, 13, 41, 107, 78, 9, 0, 2, 5, 13, 35, 115, 218, 113, 10, 0, 2, 5, 13, 34, 98, 296, 407, 157, 11, 0, 2, 5, 13, 34, 90, 276, 695, 709, 211, 12, 0, 2, 5, 13, 34, 89, 244, 750
Offset: 1

Views

Author

Clark Kimberling, Mar 30 2012

Keywords

Comments

For n>2, each row begins with 0 and ends with n+1. If the term in row n and column k is denoted by U(n,k), then U(n,n-2)=A105163(n-1).
Row sums: A000225 (-1+2^n)
Alternating row sums: A137470
Limiting row: 0,2,5,13,34,89,..., even-indexed Fibonacci numbers
For a discussion and guide to related arrays, see A208510.

Examples

			First six rows:
1
1...2
1...1...3
1...1...3...4
1...1...2...8...5
1...1...2...6...17...6
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2

Crossrefs

Cf. A210876, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + x;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210876 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210877 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

Formula

u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+x,
where u(1,x)=1, v(1,x)=1.
Showing 1-6 of 6 results.

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