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.

A133728 A128174 * A127775.

Original entry on oeis.org

1, 0, 3, 1, 0, 5, 0, 3, 0, 7, 1, 0, 5, 0, 9, 0, 3, 0, 7, 0, 11, 1, 0, 5, 0, 9, 0, 13, 0, 3, 0, 7, 0, 11, 0, 15, 1, 0, 5, 0, 9, 0, 13, 0, 17, 0, 3, 0, 7, 0, 11, 0, 15, 0, 19, 1, 0, 5, 0, 9, 0, 13, 0, 17, 0, 21, 0, 3, 0, 7, 0, 11, 0, 15, 0, 19, 0, 23
Offset: 1

Views

Author

Gary W. Adamson, Sep 21 2007

Keywords

Comments

Row sums are the triangular numbers 1, 3, 6, 10, 15, 21, 28, ...; see A000217.

Examples

			From _Philippe Deléham_, Oct 28 2011: (Start)
Triangle begins:
  1;
  0,  3;
  1,  0,  5;
  0,  3,  0,  7;
  1,  0,  5,  0,  9;
  0,  3,  0,  7,  0, 11;
  1,  0,  5,  0,  9,  0, 13; ...
(End)

Crossrefs

Cf. A000217.

Formula

A128174 * A127775 as infinite lower triangular matrices. Triangle by columns: (2k-1, 0, 2k-1, 0, ...) in k-th column.

Extensions

Corrected and extended by Philippe Deléham, Oct 28 2011

A143218 Triangle read by rows, A127775 * A000012 * A127775; 1<=k<=n.

Original entry on oeis.org

1, 3, 9, 5, 15, 25, 7, 21, 35, 49, 9, 27, 45, 63, 81, 11, 33, 55, 77, 99, 121, 13, 39, 65, 91, 117, 143, 169, 15, 45, 75, 105, 135, 165, 195, 225, 17, 51, 85, 119, 153, 187, 221, 255, 289, 19, 57, 95, 133, 171, 209, 247, 285, 323, 361, 21, 63, 105, 147, 189, 231, 273, 315, 357, 399, 441
Offset: 1

Views

Author

Gary W. Adamson, Jul 30 2008

Keywords

Examples

			First few rows of the triangle =
   1;
   3,  9;
   5, 15, 25;
   7, 21, 35, 49;
   9, 27, 45, 63,  81;
  11, 33, 55, 77,  99, 121;
  13, 39, 65, 91, 117, 143, 169;
  ...
T(5,3) = 45 = 9*5 = (2*5 - 1) * (2*3 - 1).

Links

Crossrefs

Cf. A000012, A005408, A014634, A015237 (row sums).
Cf. A016754, A024598, A033567, A127775, A131507.

Programs

  • Magma
    [(2*n-1)*(2*k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 12 2022
  • Mathematica
    Table[(2*k-1)*(2*n-1), {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 12 2022 *)
  • SageMath
    flatten([[(2*n-1)*(2*k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 12 2022

Formula

Triangle read by rows, A127775 * A000012 * A127775.
T(n, k) = (2*n - 1) * (2*k - 1), 1<=k<=n.
Sum_{k=1..n} T(n, k) = A015237(n) = n^2 * (2*n-1).
From G. C. Greubel, Jul 12 2022: (Start)
T(n, k) = A131507(n,k) * A127775(n,k).
T(n, n) = A016754(n-1) = (2*n-1)^2, n >= 1.
T(2*n-1, n) = A014634(n-1), n >= 1.
T(2*n-2, n-1) = A033567(n-1), n >= 2.
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024598(n), n >= 1. (End)

A127780 A127775 * A002260 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 6, 5, 10, 15, 7, 14, 21, 28, 9, 18, 27, 36, 45, 11, 22, 33, 44, 55, 66, 13, 26, 39, 52, 65, 78, 91, 15, 30, 45, 60, 75, 90, 105, 120
Offset: 1

Views

Author

Gary W. Adamson, Jan 28 2007

Keywords

Comments

Odd number transform of A002260.
Row sums = A002414: (1, 9, 30, 70, 135, ...).

Examples

			First few rows of the triangle:
   1;
   3,  6;
   5, 10, 15;
   7, 14, 21, 28;
   9, 18, 27, 36, 45;
  11, 22, 33, 44, 55, 66;
  ...

Crossrefs

Cf. A127775, A002260, A002414.

A134560 Triangle A051731 * A127775 (as infinite lower triangular matrices).

Original entry on oeis.org

1, 1, 3, 1, 0, 5, 1, 3, 0, 7, 1, 0, 0, 0, 9, 1, 3, 5, 0, 0, 11, 1, 0, 0, 0, 0, 0, 13, 1, 3, 0, 7, 0, 0, 0, 15, 1, 0, 5, 0, 0, 0, 0, 0, 17, 1, 3, 0, 0, 9, 0, 0, 0, 0, 19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 1, 3, 5, 7, 0, 11, 0, 0, 0, 0, 0, 23
Offset: 1

Views

Author

Gary W. Adamson, Oct 31 2007

Keywords

Comments

Obtained also by replacing the 1's in column k of A051731 with (2k-1).

Examples

			First few rows of the triangle:
  1;
  1,  3;
  1,  0,  5;
  1,  3,  0,  7;
  1,  0,  0,  0,  9;
  1,  3,  5,  0,  0, 11;
  1,  0,  0,  0,  0,  0, 13;
  1,  3,  0,  7,  0,  0,  0, 15;
  ...

Crossrefs

Cf. A129235 (row sums), A051731, A127775.

A143468 Triangle read by rows, A054525 * A127775, 1<=k<=n.

Original entry on oeis.org

1, -1, 3, -1, 0, 5, 0, -3, 0, 7, -1, 0, 0, 0, 9, 1, -3, -5, 0, 0, 11, -1, 0, 0, 0, 0, 0, 13, 0, 0, 0, -7, 0, 0, 0, 15, 0, 0, -5, 0, 0, 0, 0, 0, 17, 1, -3, 0, 0, -9, 0, 0, 0, 0, 19, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21
Offset: 1

Views

Author

Gary W. Adamson, Aug 17 2008

Keywords

Comments

Row sums = A140434: (1, 2, 4, 4, 8, 4, 12, 8, 12,...).
Left border = mu(n), A008683.

Examples

			First few rows of the triangle =
1;
-1, 3;
-1, 0, 5;
0, -3, 0, 7;
-1, 0, 0, 0, 9;
1, -3, -5, 0, 0, 11;
-1, 0, 0, 0, 0, 0, 13;
...

Crossrefs

Cf. A127775, A054525, A140434, A008683.

Formula

Triangle read by rows, A054525 * A127775, 1<=k<=n. Mobius transform of an infinite lower triangular matrix with (1, 3, 5, 7,...) in the main diagonal and the rest zeros.

A131421 Triangle read by rows (n>=1, 1<=k<=n): T(n,k) = 2*(n+k) - 3.

Original entry on oeis.org

1, 3, 5, 5, 7, 9, 7, 9, 11, 13, 9, 11, 13, 15, 17, 11, 13, 15, 17, 19, 21, 13, 15, 17, 19, 21, 23, 25, 15, 17, 19, 21, 23, 25, 27, 29, 17, 19, 21, 23, 25, 27, 29, 31, 33, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41
Offset: 1

Views

Author

Gary W. Adamson, Jul 10 2007

Keywords

Comments

Row sums = A000567, the octagonal numbers: (1, 8, 21, 40, 65, 96,...).

Examples

			First few rows of the triangle are:
1;
3, 5;
5, 7, 9;
7, 9, 11, 13;
9, 11, 13, 15, 17;
11, 13, 15, 17, 19, 21;
13, 15, 17, 19, 21, 23, 25;
...

Links

Crossrefs

Cf. A127775, A000567.

Programs

  • Mathematica
    Table[2 (n + k) - 3, {n, 150}, {k, n}] // Flatten (* Michael De Vlieger, Oct 06 2017 *)
  • PARI
    tabl(nn) = {ma = matrix(nn, nn, n, k, (k<=n)); mb = matrix(nn, nn, n, k, (2*n - 1)*(k==n)); mr = ma*mb + mb*ma - ma; for (n = 1, nn, for (k = 1, n, print1(mr[n, k], ", ");); print(););} \\ Michel Marcus, Mar 04 2014

Formula

(A000012 * A127775) + (A127775 * A000012) - A000012, where all the sequences and the result are interpreted as infinite lower triangular matrices.

Extensions

Corrected and extended by Michel Marcus, Mar 04 2014
New name from Andrey Zabolotskiy, Oct 06 2017
Showing 1-6 of 6 results.

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