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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A131302 2*A122203 - A000012.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 7, 3, 3, 1, 9, 13, 5, 3, 1, 11, 15, 7, 5, 3, 1, 13, 11, 11, 9, 5, 3, 1, 15, 9, 9, 7, 9, 5, 3, 1, 17, 7, 13, 11, 11, 11, 5, 3, 1, 19, 33, 15, 13, 7, 9, 11, 5, 3, 1
Offset: 1

Views

Author

Gary W. Adamson, Jun 27 2007

Keywords

Examples

			Row 4 = (7, 3, 3, 1) = 2*(4, 2, 2, 1) - (1, 1, 1, 1). First few rows of the triangle are:
1;
3, 1;
5, 5, 1;
7, 3, 3, 1;
9, 13, 5, 3, 1;
11, 15, 7, 5, 3, 1;
13, 11, 11 9, 5, 3, 1;
...

Crossrefs

Cf. A122203, A000012.

Formula

2*A122203(deleting the right border of zeros) - A000012.

A131324 2*A049310 - A000012(signed).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 11, 1, 9, 1, 1, 1, 7, 1, 19, 1, 11, 1, 1, 1, 1, 19, 1, 29, 1, 13, 1, 1, 1, 9, 1, 39, 1, 41, 1, 15, 1, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 28 2007

Keywords

Comments

Row sums = A062114: (1, 2, 3, 6, 9, 16, 25, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  1,  5,  1,  1;
  1,  5,  1,  7,  1,  1,
  1,  1, 11,  1,  9,  1,  1;
  1,  7,  1, 19,  1, 11,  1,  1;
  ...

Crossrefs

Cf. A049310, A062114, A131325, A131326, A131327.

Formula

2*A049310 - A000012(signed + - + - ... by columns).

A131402 2*A007318 - (A046854 + A065941 - A000012).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 7, 6, 1, 1, 7, 14, 14, 7, 1, 1, 9, 20, 33, 20, 9, 1, 1, 10, 31, 56, 56, 31, 10, 1, 1, 12, 40, 97, 111, 97, 40, 12, 1, 1, 13, 55, 142, 217, 217, 142, 55, 13, 1, 1, 15, 67, 213, 358, 463, 358, 213, 67, 15, 1, 1, 16, 86, 287, 590, 841, 841, 590, 287, 86, 16, 1
Offset: 0

Views

Author

Gary W. Adamson, Jul 07 2007

Keywords

Comments

Row sums = A131403: (1, 2, 5, 10, 21, 44, 93, ...).

Examples

			First few rows of the triangle are:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  4,  1;
  1,  6,  7,  6,  1;
  1,  7, 14, 14,  7,  1;
  1,  9, 20, 33, 20,  9,  1;
  1, 10, 31, 56, 56, 31, 10,  1;
  ...

Links

Crossrefs

Row sums are A131403.
Cf. A046854, A065941.

Programs

  • PARI
    T(n,k) = if(k <= n, 2*binomial(n, k) + 1 - binomial((n + k)\2, k) - binomial(n-(k+1)\2, k\2), 0) \\ Andrew Howroyd, Aug 09 2018

Formula

2*A007318 - (A046854 + A065941 - A000012) as infinite lower triangular matrices.
T(n,k) = 2*binomial(n, k) + 1 - binomial(floor((n + k)/2), k) - binomial(n-floor((k+1)/2), floor(k/2)). - Andrew Howroyd, Aug 09 2018

Extensions

Missing terms inserted and a(55) and beyond from Andrew Howroyd, Aug 09 2018

A131406 3*A128174 - 2*A000012(signed).

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Jul 07 2007

Keywords

Comments

Row sums = A032766, congruent to {0,1} mod 3: (1, 3, 4, 6, 7, 9, 10,...).
Sequence array for the expansion of (1+2x)/(1-x^2). A105476 is an eigensequence. [From Paul Barry, Nov 03 2010]

Examples

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

Crossrefs

Cf. A128174, A000012, A032766.

Programs

  • Mathematica
    T[n_, k_] := Mod[n-k, 2]+1; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2016 *)

Formula

3*A128174 - 2*A000012(signed + - + 1 by columns). (1, 2, 1, 2, 1,...) in every column.
Triangle T(n,k)=if(k<=n,(3-(-1)^(n-k))/2). [From Paul Barry, Nov 03 2010]

A132307 2*A007318^(2) - A000012.

Original entry on oeis.org

1, 3, 1, 7, 7, 1, 15, 23, 11, 1, 31, 63, 47, 15, 1, 63, 159, 159, 79, 19, 1, 127, 383, 479, 319, 119, 23, 1, 255, 895, 1343, 1119, 559, 167, 27, 1, 511, 2047, 3583, 3583, 2239, 895, 223, 31, 1, 1023, 4607, 9215, 10751, 8063, 4031, 1343, 287, 35, 1
Offset: 0

Views

Author

Gary W. Adamson, Aug 18 2007

Keywords

Comments

Row sums = A132308: (1, 4, 15, 50, 157, 480, 1451, ...). Inverse binomial transform of A132307 = triangle A132309 (having row sums A077552).

Examples

			First few rows of the triangle:
   1;
   3,   1;
   7,   7,   1;
  15,  23,  11,  1;
  31,  63,  47, 15,  1;
  63, 159, 159, 79, 19, 1;
  ...

Crossrefs

Cf. A132308, A132309, A077552.

Formula

2*A007318^(2) - A000012 as infinite lower triangular matrices.

A133735 A000012 * A133734.

Original entry on oeis.org

1, 2, 0, 3, 0, 1, 4, 0, 2, 1, 5, 0, 3, 2, 2, 6, 0, 4, 3, 4, 2, 7, 0, 5, 4, 6, 4, 4, 8, 0, 6, 5, 8, 6, 8, 4, 9, 0, 7, 6, 10, 8, 12, 8, 7, 10, 0, 8, 7, 12, 10, 16, 12, 14, 8
Offset: 0

Views

Author

Gary W. Adamson, Sep 22 2007

Keywords

Comments

Row sums = A000070: (1, 2, 4, 7, 12, 19, 30, 45, ...).
Right border = A002865: (1, 0, 1, 1, 2, 2, 4, 4, 7, 8, ...).

Examples

			First few rows of the triangle:
  1;
  2, 0;
  3, 0, 1;
  4, 0, 2, 1;
  5, 0, 3, 2,  2;
  6, 0, 4, 3,  4, 2;
  7, 0, 5, 4,  6, 4,  4;
  8, 0, 6, 5,  8, 6,  8, 4;
  9, 0, 7, 6, 10, 8, 12, 8, 7;
  ...

Crossrefs

Cf. A133734, A000070.

Formula

A000012 * A133734 as infinite lower triangular matrices.

A134540 A054525 * A000012.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Oct 31 2007

Keywords

Comments

Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, 4, 6, 4, ...).

Examples

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

Crossrefs

Cf. A054525.

Formula

A054525 * A000012 as infinite lower triangular matrices.
Triangle read by rows, partial sums of A054525 (the Mobius transform) terms starting from the right.

A134542 A134541 * A000012.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 2, 3, 3, 2, 1, 1, 3, 4, 4, 3, 2, 1, 1, 3, 4, 4, 4, 3, 2, 1, 1, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 4, 5, 5, 5, 4, 3, 2, 1
Offset: 1

Views

Author

Gary W. Adamson, Oct 31 2007

Keywords

Comments

Row sums = A002088: (1, 2, 4, 6, 10, 12, 18, 22, ...).
A134542 * [1,2,3,...] = A015631: (1, 3, 8, 15, 29, 42, ...).

Examples

			First few rows of the triangle:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 2, 1;
  1, 3, 3, 2, 1;
  1, 2, 3, 3, 2, 1;
  1, 3, 4, 4, 3, 2, 1;
  1, 3, 4, 4, 4, 3, 2, 1;
  1, 3, 4, 5, 5, 4, 3, 2, 1;
  1, 2, 4, 5, 5, 5, 4, 3, 2, 1;
  ...

Crossrefs

Cf. A134541, A002088, A015631.

Formula

A134541 * A000012 as infinite lower triangular matrices.
Triangle read by rows, partial sums of A134541 terms starting from the right.

A135841 A000012 * A135839 as infinite lower triangular matrices.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Dec 01 2007

Keywords

Comments

Row sums = A024206: (1, 3, 5, 8, 11, 15, 19, ...).

Examples

			First few rows of the triangle:
  1;
  2, 1;
  3, 1, 1;
  4, 2, 1, 1;
  5, 2, 2, 1, 1;
  6, 3, 2, 2, 1, 1;
  7, 3, 3, 2, 2, 1, 1;
  ...

Links

Crossrefs

Cf. A135839, A024206.

Programs

  • Mathematica
    T[1, 1] := 1; T[n_, 1] := n; T[n_, n_] := 1; T[n_, k_] := Floor[(n - k + 2)/2]; Table[T[n, k], {n, 1, 15}, {k, 1, n}]//Flatten (* G. C. Greubel, Dec 06 2016 *)

Formula

T(1, 1) = 1, T(n, 1) = n, T(n, n) = 1, T(n, k) = floor((n - k + 2)/2). - G. C. Greubel, Dec 06 2016

Extensions

Terms a(56) and beyond from G. C. Greubel, Dec 06 2016

A143595 Triangle read by rows, A000012 * (an infinite lower triangular matrix with 1's in the first column and the rest 2's); 1<=k<=n.

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 4, 6, 4, 2, 5, 8, 6, 4, 2, 6, 10, 8, 6, 4, 2, 7, 12, 10, 8, 6, 4, 2, 8, 14, 12, 10, 8, 6, 4, 2, 9, 16, 14, 12, 10, 8, 6, 4, 2, 10, 18, 16, 14, 12, 10, 8, 6, 4, 2, 11, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2
Offset: 1

Views

Author

Gary W. Adamson, Aug 26 2008

Keywords

Comments

Row sums = n^2.
The linear sequence A056944 is more appropriately a triangle, (reversal of A143595).

Examples

			First few rows of the triangle =
1;
2, 2;
3, 4, 2;
4, 6, 4, 2;
5, 8, 6, 4, 2;
6, 10, 8, 6, 4, 2;
7, 12, 10, 8, 6, 4, 2;
...

Crossrefs

Cf. A056944.

Formula

Triangle read by rows, A000012 * (an infinite lower triangular matrix with 1's in the first column and the rest 2's); i.e. (1; 1,2; 1,2,2;...). A000012 = an infinite lower triangular matrix with all 1's. 1<=k<=n
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