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 1-9 of 9 results.

A077934 Duplicate of A077973.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 3, 3, 13, 19, 13, 13, 51, 77, 51, 51, 205, 307, 205, 205, 819, 1229, 819, 819
Offset: 0

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Author

Keywords

A077950 Expansion of 1/(1-x+2*x^3).

Original entry on oeis.org

1, 1, 1, -1, -3, -5, -3, 3, 13, 19, 13, -13, -51, -77, -51, 51, 205, 307, 205, -205, -819, -1229, -819, 819, 3277, 4915, 3277, -3277, -13107, -19661, -13107, 13107, 52429, 78643, 52429, -52429, -209715, -314573, -209715, 209715, 838861, 1258291, 838861, -838861, -3355443, -5033165, -3355443
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002, Jun 17 2007

Keywords

Links

Crossrefs

Cf. A077973.

Programs

  • GAP
    a:=[1,1,1];; for n in [4..50] do a[n]:= a[n-1]-2*a[n-3]; od; a; # G. C. Greubel, Jul 03 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x+2*x^3) )); // G. C. Greubel, Jul 03 2019
  • Mathematica
    CoefficientList[Series[1/(1-x+2x^3),{x,0,50}],x] (* or *) LinearRecurrence[ {1,0,-2}, {1,1,1},50] (* Harvey P. Dale, Oct 18 2013 *)
  • PARI
    Vec(1/(1-x+2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 26 2012
  • Sage
    (1/(1-x+2*x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Jul 03 2019

Formula

a(n) = a(n-1) - 2*a(n-3), where a(0)=1, a(1)=1, a(2)=1. - Sergei N. Gladkovskii, Aug 21 2012
a(n) = (-1)^n * A077973(n). - Ralf Stephan, Aug 18 2013
G.f.: G(0)/(2*(1-x^2)), where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
a(n) = Sum_{k=1..floor((n+2)/2)} binomial(n+2-2*k, k-1)*(-2)^(k-1). - Taras Goy, Sep 18 2019

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.

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.

A137505 Inverse binomial transform of A007910.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 4, -4, 4, 4, -12, 20, -12, -12, 52, -76, 52, 52, -204, 308, -204, -204, 820, -1228, 820, 820, -3276, 4916, -3276, -3276, 13108, -19660, 13108, 13108, -52428, 78644, -52428, -52428, 209716, -314572, 209716, 209716, -838860, 1258292, -838860, -838860, 3355444, -5033164, 3355444
Offset: 0

Views

Author

Paul Curtz, Apr 23 2008

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{-1,0,2},{1,1,0},50] (* Harvey P. Dale, Sep 17 2012 *)

Formula

Recurrence: a(n) = -a(n-1) + 2a(n-3), starting 1,1,0.
O.g.f.: (1+x)^2/((1-x)(1+2x+2x^2)). - R. J. Mathar, Jun 12 2008
a(4n) = a(4n+1) = (-1)^n*A109499(n). - Paul Curtz, Nov 01 2009
a(n) = (1/5) * (A137429(n-1) + 4) = A077973(n-2) + 2*A077973(n-1) + A077973(n). - Ralf Stephan, Aug 18 2013

Extensions

More terms from R. J. Mathar, Jun 12 2008

A137500 Binomial transform of b(n) = (0, 0, A007910).

Original entry on oeis.org

0, 0, 1, 5, 17, 51, 149, 439, 1309, 3927, 11797, 35423, 106301, 318903, 956645, 2869807, 8609293, 25827879, 77483893, 232452191, 697357085, 2092071255, 6276212741, 18828636175, 56485906477, 169457719431, 508373162389, 1525119495359, 4575358494269, 13726075482807
Offset: 0

Views

Author

Paul Curtz, Apr 27 2008

Keywords

Comments

b(n) is binomial transform of (0, 0, A077973).

Links

Crossrefs

Cf. A007910, A009545.

Programs

  • Mathematica
    LinearRecurrence[{5,-8,6},{0,0,1},40] (* Harvey P. Dale, Sep 27 2020 *)
  • PARI
    concat([0,0], Vec(1/((1 - 3*x)*(1 - 2*x + 2*x^2)) + O(x^40))) \\ Andrew Howroyd, Jan 03 2020

Formula

a(n) = 3*a(n-1) + A009545(n-1) for n > 0.
From Andrew Howroyd, Jan 03 2020: (Start)
a(n) = Sum_{k=0..n-2} binomial(n, k+2)*A007910(k).
a(n) = 5*a(n-1) - 8*a(n-2) + 6*a(n-3) for n >= 3.
G.f.: x*2/((1 - 3*x)*(1 - 2*x + 2*x^2)). (End)

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 03 2020

A317505 Triangle read by rows: T(0,0) = 1; T(n,k) = - T(n-1,k) - 2 T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, -1, 1, -1, 2, 1, -4, -1, 6, 1, -8, 4, -1, 10, -12, 1, -12, 24, -1, 14, -40, 8, 1, -16, 60, -32, -1, 18, -84, 80, 1, -20, 112, -160, 16, -1, 22, -144, 280, -80, 1, -24, 180, -448, 240, -1, 26, -220, 672, -560, 32, 1, -28, 264, -960, 1120, -192, -1, 30, -312, 1320, -2016, 672, 1, -32, 364, -1760, 3360, -1792, 64, -1, 34, -420, 2288, -5280, 4032, -448
Offset: 0

Views

Author

Shara Lalo, Aug 02 2018

Keywords

Comments

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1+x+2x^3) are given by the sequence generated by the row sums (see A077973).

Examples

			Triangle begins:
   1;
  -1;
   1;
  -1,   2;
   1,  -4;
  -1,   6;
   1,  -8,    4;
  -1,  10,  -12;
   1, -12,   24;
  -1,  14,  -40,     8;
   1, -16,   60,   -32;
  -1,  18,  -84,    80;
   1, -20,  112,  -160,    16;
  -1,  22, -144,   280,   -80;
   1, -24,  180,  -448,   240;
  -1,  26, -220,   672,  -560,    32;
   1, -28,  264,  -960,  1120,  -192;
  -1,  30, -312,  1320, -2016,   672;
   1, -32,  364, -1760,  3360, -1792,   64;
  -1,  34, -420,  2288, -5280,  4032, -448;

References

  • Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.

Links

Crossrefs

Row sums give A077973.
Cf. A065109, A303872.
Cf. A133156, A305098.

Programs

  • Mathematica
    t[n_, k_] := t[n, k] = (-1)^(n - 3k) * 2^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, - t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]}] // Flatten

Formula

T(n,k) = (-1)^(n - 3k) * 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).
Showing 1-9 of 9 results.

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