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.

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A130459 A059268 * A097806.

Original entry on oeis.org

1, 3, 2, 3, 6, 4, 3, 6, 12, 8, 3, 6, 12, 24, 16, 3, 6, 12, 24, 48, 32, 3, 6, 12, 24, 48, 96, 64, 3, 6, 12, 24, 48, 96, 192, 128, 3, 6, 12, 24, 48, 96, 192, 384, 256
Offset: 1

Views

Author

Gary W. Adamson, May 26 2007

Keywords

Comments

Row sums = A036563 starting (1, 5, 13, 29, 61, 125, ...).

Examples

			First few rows of the triangle:
  1;
  3, 2;
  3, 6,  4;
  3, 6, 12,  8;
  3, 6, 12, 24, 16;
  3, 6, 12, 24, 48, 32;
  ...

Crossrefs

Cf. A059268, A097806, A036563.

Formula

A059268 * A097806 as infinite lower triangular matrices. A059268 = [1; 1,2; 1,2,4; ...]. A097806 = the pairwise operator.

A131032 A097806 * A130296.

Original entry on oeis.org

1, 3, 1, 5, 2, 1, 7, 2, 2, 1, 9, 2, 2, 2, 1, 11, 2, 2, 2, 2, 1, 13, 2, 2, 2, 2, 2, 1, 15, 2, 2, 2, 2, 2, 2, 1, 17, 2, 2, 2, 2, 2, 2, 2, 1, 19, 2, 2, 2, 2, 2, 2, 2, 2, 1, 21, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 23, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1
Offset: 1

Views

Author

Gary W. Adamson, Jun 10 2007

Keywords

Comments

Row sums give A008574.
A131033 = A130296 * A097806.

Examples

			First few rows of the triangle are:
1;
3, 1;
5, 2, 1;
7, 2, 2, 1;
9, 2, 2, 2, 1;
11, 2, 2, 2, 2, 1;
13, 2, 2, 2, 2, 2, 1;
...

Crossrefs

Cf. A008574, A097806, A130296, A131033.

Formula

A097806 * A130296 as infinite lower triangular matrices. A097806 = the pairwise operator, A130296 = (1; 2,1; 3,1,1; ...).

Extensions

Definition corrected and more terms from Georg Fischer, Oct 10 2021

A131129 3*A007318 - 2*A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 3, 9, 7, 1, 3, 12, 18, 10, 1, 3, 15, 30, 30, 13, 1, 3, 18, 45, 60, 45, 16, 1, 3, 21, 63, 105, 105, 63, 19, 1, 3, 24, 84, 168, 210, 168, 84, 22, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 16 2007

Keywords

Comments

Row sums = A131128: (1, 2, 8, 20, 44, 92, 188, 380, ...), the binomial transform of (1, 1, 5, 1, 5, 1, 5, ...). Triangle A131108 has row sums (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...). Generalization: Given triangles generated from N*A007318 - (N-1)*A097806, row sums are binomial transforms of (1, 1, (2N-1), 1, (2N-1), 1, ...).
Triangle T(n,k), 0 <= k <= n, read by rows given by [1,2,-3,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

Examples

			First few rows of the triangle:
  1;
  1,  1;
  3,  4,  1;
  3,  9,  7,  1;
  3, 12, 18, 10,  1;
  3, 15, 30, 30, 13,  1;
  ...

Crossrefs

Cf. A097806, A131128, A095121.

Formula

G.f.: (1-x*y+2*x^2+2*x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015

A133093 A007318 * A097806 * A133080.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 10, 6, 5, 1, 15, 10, 15, 5, 1, 21, 15, 35, 15, 7, 1, 28, 21, 70, 35, 28, 7, 1, 36, 28, 126, 70, 84, 28, 9, 1, 45, 36, 210, 126, 210, 84, 45, 9, 1, 55, 45, 330, 210, 462, 210, 165, 45, 11, 1
Offset: 1

Views

Author

Gary W. Adamson, Sep 09 2007

Keywords

Comments

Row sums = A033484: (1, 4, 10, 22, 46, 94, ...).

Examples

			First few rows of the triangle:
   1;
   3,  1;
   6,  3,  1;
  10,  6,  5,  1;
  15, 10, 15,  5,  1;
  21, 15, 35, 15,  7,  1;
  28, 21, 70, 35, 28,  7,  1;
  ...

Crossrefs

Cf. A133080, A097806, A033484. Duplicate of A131110.

Formula

A007318 * A097806 * A133080 as infinite lower triangular matrices.
Binomial transform of an infinite lower triangular matrix with (1,1,1,...) in the main diagonal, (2,1,2,1,...) in the subdiagonal and (1,0,1,0,...) in the subsubdiagonal.

A126705 A097806 * A054523 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 1, 1, 6, 1, 0, 1, 1, 6, 2, 1, 0, 1, 1, 8, 2, 1, 0, 0, 1, 1, 10, 2, 0, 1, 0, 0, 1, 1, 10, 2, 2, 1, 0, 0, 0, 1, 1, 10, 4, 2, 0, 1, 0, 0, 0, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, Apr 17 2007

Keywords

Comments

Row sums = (1, 3, 5, 7, 9, ...). A129479 = A054523 * A097806. A097806 = the pairwise operator.

Examples

			First few rows of the triangle:
  1;
  2, 1;
  3, 1, 1;
  4, 1, 1, 1;
  6, 1, 0, 1, 1;
  6, 2, 1, 0, 1, 1;
  8, 2, 1, 0, 0, 1, 1;
  ...

Crossrefs

Cf. A097806 * A054523, A129479.

A128184 A051731 * A097806.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 0, 0, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, Feb 17 2007

Keywords

Comments

Row sums = A114003: (1, 3, 3, 5, 3, 7, 3, 7, 5, 7, ...).

Examples

			First few rows of the triangle:
  1;
  2, 1;
  1, 1, 1;
  2, 1, 1, 1;
  1, 0, 0, 1, 1;
  2, 2, 1, 0, 1, 1;
  ...

Crossrefs

Cf. A051731, A097806, A114003.

Formula

A051731 * A097806, (inverse Moebius transform of A097806).

A129572 A129372 * A097806.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, Apr 22 2007

Keywords

Comments

Row sums = A086374: (1, 2, 3, 2, 3, 4, 3, 2, 5, 4, ...). A129573 = A097806 * A129372.

Examples

			First few rows of the triangle:
  1;
  1, 1;
  1, 1, 1;
  0, 0, 1, 1;
  1, 0, 0, 1, 1;
  1, 1, 0, 0, 1, 1;
  1, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 1, 1;
  ...

Crossrefs

Cf. A129372, A097806, A086374, A129573.

Formula

A129372 * A097806 as infinite lower triangular matrices.

A131131 4*A007318 - 3*A097806.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 4, 12, 9, 1, 4, 16, 24, 13, 1, 4, 20, 40, 40, 17, 1, 4, 24, 60, 80, 60, 21, 1, 4, 28, 84, 140, 140, 84, 25, 1, 4, 32, 112, 224, 280, 224, 112, 29, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 16 2007

Keywords

Comments

Row sums = A131130, (1, 2, 10, 26, 52, 98, 190, ...), the binomial transform of (1, 1, 7, 1, 7, 1, ...). Generally, triangles generated from N*A007318 - (N-1)*A097806 have row sums that are binomial transforms of (1, 1, (N-1), 1, (N-1), 1, ...). A095121 = (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...) and = row sums of A131108.
Triangle T(n,k), 0 <= k <= n,read by rows given by [1,3,-4,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

Examples

			First few rows of the triangle:
  1;
  1,  1;
  4,  5,  1;
  4, 12,  9,  1;
  4, 16, 24, 13,  1
  4, 20, 40, 40, 17,  1;
  ...

Crossrefs

Cf. A131130, A131129, A131128, A131127, A046055, A131108, A095121, A097806.

Formula

4*A007318 - 3*A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.
G.f.: (1-x*y+3*x^2+3*x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015

A133094 A007318 * A133080 * A097806, as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 7, 7, 5, 1, 9, 14, 14, 5, 1, 11, 25, 30, 16, 7, 1, 13, 41, 55, 41, 27, 7, 1, 15, 63, 91, 91, 77, 29, 9, 1, 17, 92, 140, 182, 182, 92, 44, 9, 1, 19, 129, 204, 336, 378, 246, 156, 46, 11, 1
Offset: 1

Views

Author

Gary W. Adamson, Sep 09 2007

Keywords

Comments

Row sums give A133095.
Binomial transform of an infinite lower triangular matrix with (1,1,1,...) in the main diagonal, (2,1,2,1,...) in the subdiagonal and (0,1,0,1,...) in the subsubdiagonal.

Examples

			First few rows of the triangle:
   1;
   3,  1;
   5,  3,  1;
   7,  7,  5,  1;
   9, 14, 14,  5,  1;
  11, 25, 30, 16,  7,  1;
  13, 41, 55, 41, 27,  7,  1;
  ...

Crossrefs

Cf. A133080, A097806, A133095.

A133601 A007318 * (A097806 + A133080 - I), I = Identity matrix.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 7, 6, 5, 1, 9, 10, 14, 5, 1, 11, 15, 30, 15, 7, 1, 13, 21, 55, 35, 27, 7, 1, 15, 28, 91, 70, 77, 28, 9, 1, 17, 36, 140, 126, 182, 84, 44, 9, 1, 19, 45, 204, 210, 378, 210, 156, 45, 11, 1
Offset: 0

Views

Author

Gary W. Adamson, Sep 18 2007

Keywords

Examples

			First few rows of the triangle are:
1;
3, 1;
5, 3, 1;
7, 6, 5, 1;
9, 10, 14, 5, 1;
11, 15, 30, 15, 7, 1;
13, 21, 55, 35, 27, 7, 1;
15, 28, 91, 70, 77, 28, 9, 1;
...

Crossrefs

Cf. A097806, A133080, A052549 (row sums).

Programs

Formula

A007318 * (A097806 + A133080 - I), I = Identity matrix. Binomial transform of an infinite lower triangular matrix with (1,1,1,...) in the main diagonal and (2,1,2,1,2,...) in the subdiagonal; and the rest zeros.
Previous Showing 11-20 of 46 results. Next

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