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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A131053 A000012 * A131047.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 6, 3, 4, 3, 6, 13, 4, 5, 3, 12, 13, 24, 5, 6, 4, 12, 34, 24, 40, 6, 7, 4, 20, 34, 80, 40, 62, 7, 8, 5, 20, 70, 80, 166, 62, 91, 8, 9
Offset: 1

Views

Author

Gary W. Adamson, Jun 12 2007

Keywords

Comments

Row sums = (2^n - 1): (1, 3, 7, 15, ...).
Left border = A004526: (1, 1, 2, 2, 3, 3, ...).
A131052 = A131047 * A000012.

Examples

			First few rows of the triangle:
  1;
  1,  2;
  2,  2,  3;
  2,  6,  3,  4;
  3,  6, 13,  4,  5;
  3, 12, 13, 24,  5,  6;
  4, 12, 34, 24, 40,  6,  7;
  ...

Crossrefs

Cf. A131047, A000012, A131052, A004526.

Formula

A000012 * A131047 as infinite lower triangular matrices.

A131252 A052509 * A000012.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 7, 6, 3, 1, 12, 11, 7, 3, 1, 20, 19, 14, 7, 3, 1, 33, 32, 26, 15, 7, 3, 1, 54, 53, 46, 30, 15, 7, 3, 1, 88, 87, 79, 57, 31, 15, 7, 3, 1, 143, 142, 133, 104, 62, 31, 15, 7, 3, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 23 2007

Keywords

Comments

Left border = A000071, Fibonacci numbers - 1, starting with F(3): (1, 2, 4, 7, 12, ...). Row sums = A131253: (1, 3, 8, 17, 34, 64, 117, ...).

Examples

			First few rows of the triangle:
   1;
   2,  1;
   4,  3,  1;
   7,  6,  3,  1;
  12, 11,  7,  3,  1;
  20, 19, 14,  7,  3,  1;
  33, 32, 26, 15,  7,  3,  1;
  ...

Crossrefs

Cf. A000071, A000012, A052509, A131253.

Formula

A052509 * A000012, where A000012 = (1; 1,1; 1,1,1; ...).

A131303 2*A122204(deleting the right border of zeros) - A000012.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 7, 3, 3, 1, 9, 15, 5, 3, 1, 11, 13, 7, 5, 3, 1, 13, 11, 11, 9, 5, 3, 115, 7, 9, 7, 9, 5, 3, 1, 17, 9, 13, 11, 11, 11, 5, 3, 1, 19, 43, 15, 13, 7, 7, 7, 5, 3, 1
Offset: 1

Views

Author

Gary W. Adamson, Jun 27 2007

Keywords

Examples

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

Crossrefs

Cf. A122204, A000012, A122203, A131302.

A131416 (A000012 * A002260) + (A002260 * A000012) - A000012.

Original entry on oeis.org

1, 4, 3, 8, 8, 5, 13, 14, 12, 7, 19, 21, 20, 16, 9, 26, 29, 29, 26, 20, 11, 34, 38, 39, 37, 32, 24, 13, 43, 48, 50, 49, 45, 38, 28, 15, 53, 59, 62, 62, 59, 53, 44, 32, 17, 64, 71, 75, 76, 74, 69, 61, 50, 36, 19
Offset: 1

Views

Author

Gary W. Adamson, Jul 08 2007

Keywords

Comments

Left column = A034856: (1, 4, 8, 13, 19, 26, 34,...). Row sums = A127735: (1, 7, 21, 46, 85...).

Examples

			First few rows of the triangle are:
1;
4, 3;
8, 8, 5;
13, 14, 12, 7;
19, 21, 20, 16, 9;
26, 29, 29, 26, 20, 11;
...

Crossrefs

Cf. A002260, A034856, A127735, A000012.

Formula

(A000012 * A002260) + (A002260 * A000012) - A000012; as infinite lower triangular matrices.

A131432 Triangle read by rows: (A000012 * A131431) + (A131431 * A000012) - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 4, 7, 7, 10, 13, 10, 13, 16, 19, 13, 16, 19, 22, 25, 16, 19, 22, 25, 28, 31, 19, 22, 25, 28, 31, 34, 37, 22, 25, 28, 31, 34, 37, 40, 43, 25, 28, 31, 34, 37, 40, 43, 46, 49, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67
Offset: 0

Views

Author

Gary W. Adamson, Jul 10 2007

Keywords

Comments

Left border = A016777, (1, 4, 7, 10, 13, ...).
Right border = A016921: (1, 7, 13, 19, 25, ...).
Row sums = A051682, 11-gonal numbers: (1, 11, 30, 58, 95, ...).

Examples

			First few rows of the triangle:
   1;
   4,  7;
   7, 10, 13;
  10, 13, 16, 19;
  13, 16, 19, 22, 25;
  16, 19, 22, 25, 28, 31;
  19, 22, 25, 28, 31, 34, 37;
  ...

Crossrefs

Cf. A016777, A051682, A131431, A016921.

Extensions

a(28) ff. corrected and more terms from Georg Fischer, Jun 05 2023

A131768 2*(A007318 * A097807) - A000012.

Original entry on oeis.org

1, 3, 1, 5, 5, 1, 7, 11, 7, 1, 9, 19, 19, 9, 1, 11, 29, 39, 29, 11, 1, 13, 41, 69, 69, 41, 13, 1, 15, 55, 111, 139, 111, 55, 15, 1, 17, 71, 167, 251, 251, 167, 71, 17, 1, 19, 89, 239, 419, 503, 419, 239, 89, 19, 1
Offset: 0

Views

Author

Gary W. Adamson, Jul 13 2007

Keywords

Comments

Row sums = A000295, the Eulerian numbers, starting (1, 4, 11, 26, 57, 120, ...).

Examples

			First few rows of the triangle:
   1;
   3,  1;
   5,  5,   1;
   7, 11,   7,   1;
   9, 19,  19,   9,   1;
  11, 29,  39,  29,  11,  1;
  13, 41,  69,  69,  41, 13,  1;
  15, 55, 111, 139, 111, 55, 15, 1;
  ...

Crossrefs

Cf. A007318, A097807, A000012, A000295.

Formula

2*(A007318 * A097807) - A000012 as infinite lower triangular matrices.

A131875 Triangle, A000012 * A131844 as infinite lower triangular matrices.

Original entry on oeis.org

1, 5, 4, 12, 5, 7, 22, 6, 8, 10, 35, 7, 9, 11, 13, 51, 8, 10, 12, 14, 16, 70, 9, 11, 13, 15, 17, 19, 92, 10, 12, 14, 16, 18, 20, 22, 117, 11, 13, 15, 17, 19, 21, 23, 25, 145, 12, 14, 16, 18, 20, 22, 24, 26, 28
Offset: 0

Views

Author

Gary W. Adamson, Jul 22 2007

Keywords

Comments

Left column = pentagonal numbers, A000326: (1, 5, 12, 22, ...).
Row sums = A001106: (1, 9, 24, 46, 75, 111, ...).

Examples

			First few rows of the triangle:
   1;
   5,  4;
  12,  5,  7;
  22,  6,  8, 10;
  35,  7,  9, 11, 13;
  51,  8, 10, 12, 14, 16;
  70,  9, 11, 13, 15, 17, 19;
  92, 10, 12, 14, 16, 18, 20, 22;
  ...

Crossrefs

Cf. A131844, A001106, A000326.

A131894 A131843 * A000012.

Original entry on oeis.org

1, 6, 3, 11, 6, 5, 16, 9, 8, 7, 21, 12, 11, 10, 9, 26, 15, 14, 13, 12, 11, 31, 18, 17, 16, 15, 14, 13, 36, 21, 20, 19, 18, 17, 16, 15, 41, 24, 23, 22, 21, 20, 19, 18, 17, 46, 27, 26, 25, 24, 23, 22, 21, 20, 19
Offset: 0

Views

Author

Gary W. Adamson, Jul 24 2007

Keywords

Comments

Left column = A016861, 5n + 1: (1, 6, 11, 16, 21, 26, ...).
Row sums = A131895: (1, 9, 22, 40, 63, ...).
Triangle A131896 = A000012 * A131843.

Examples

			First few rows of the triangle:
   1;
   6,  3;
  11,  6,  5;
  16,  9,  8,  7;
  21, 12, 11, 10,  9;
  26, 15, 14, 13, 12, 11;
  ...

Crossrefs

Cf. A131843, A131895, A131896.

Formula

Triangle read by rows, partial row sums starting from the right, of A131843.

A131896 A000012 * A131843.

Original entry on oeis.org

1, 4, 3, 9, 4, 5, 16, 5, 6, 7, 25, 6, 7, 8, 9, 36, 7, 8, 9, 10, 11, 49, 8, 9, 10, 11, 12, 13, 64, 9, 10, 11, 12, 13, 14, 15, 81, 10, 11, 12, 13, 14, 15, 16, 17, 100, 11, 12, 13, 14, 15, 16, 17, 18, 19
Offset: 1

Views

Author

Gary W. Adamson, Jul 24 2007

Keywords

Comments

Row sums = A000566: (1, 7, 18, 34, 55, 81, ...), binomial transform of (1, 6, 5, 0, 0, 0, ...).
A131894 = A131843 * A000012.

Examples

			First few rows of the triangle:
   1;
   4,  3;
   9,  4,  5;
  16,  5,  6,  7;
  25,  6,  7,  8,  9;
  36,  7,  8,  9, 10, 11;
  49,  8,  9, 10, 11, 12, 13;
  64,  9, 10, 11, 12, 13, 14, 15;
  ...

Crossrefs

Cf. A131843, A000566.

Formula

A000012 * A131843 as infinite lower triangular matrices.

A133083 A000012 * A133080.

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Sep 08 2007

Keywords

Comments

Row sums = A032766, congruent to {0,1} (mod 3): (1, 3, 4, 6, 7, 9, 10, ...).

Examples

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

Links

Crossrefs

Cf. A133080, A000012, A032766.

Programs

  • Mathematica
    T[n_, k_] := If[k == n, 1, 1  + (1 - (-1)^k)/2 ]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)
  • PARI
    for(n=1,10, for(k=1,n, print1(if(k==n, 1, 1 + (1-(-1)^k)/2), ", "))) \\ G. C. Greubel, Oct 21 2017

Formula

A000012 * A133080 as infinite lower triangular matrices.
Previous Showing 51-60 of 2635 results. Next

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