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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A133115 Triangle read by rows: 2*A133113 - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 7, 3, 1, 11, 7, 7, 1, 17, 11, 21, 7, 1, 23, 17, 47, 21, 11, 1, 31, 23, 91, 47, 43, 11, 1, 39, 31, 159, 91, 123, 43, 15, 1, 49, 39, 259, 159, 295, 123, 73, 15, 1, 59, 49, 399, 259, 627, 295, 255, 73, 19, 1, 71, 59, 589, 399, 1219, 627, 733, 255, 111, 19, 1, 83, 71, 839, 589, 2211, 1219, 1839, 733, 459, 111, 23, 1
Offset: 1

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Author

Gary W. Adamson, Sep 14 2007

Keywords

Comments

Left column = A047838: (1, 3, 7, 11, 17, 23, ...).
Row sums = A000295: (1, 4, 11, 26, 57, ...).

Examples

			First few rows of the triangle:
   1;
   3,  1;
   7,  3,  1;
  11,  7,  7,  1;
  17, 11, 21,  7,  1;
  23, 17, 47, 21, 11,  1;
  31, 23, 91, 47, 43, 11,  1;
  ...

Crossrefs

Cf. A000295, A047838, A133113.

Extensions

a(46) = 59 corrected and more terms from Georg Fischer, Jun 10 2023

A376182 Triangle T read by rows: T(n, k) = (2*n^2 + 4*n + 1 - (-1)^n) / 4 - (1 + (-1)^k) * (n - k) - k.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 11, 6, 9, 8, 17, 10, 15, 12, 13, 23, 14, 21, 16, 19, 18, 31, 20, 29, 22, 27, 24, 25, 39, 26, 37, 28, 35, 30, 33, 32, 49, 34, 47, 36, 45, 38, 43, 40, 41, 59, 42, 57, 44, 55, 46, 53, 48, 51, 50, 71, 52, 69, 54, 67, 56, 65, 58, 63, 60, 61, 83, 62, 81, 64, 79, 66, 77, 68, 75, 70, 73, 72
Offset: 1

Views

Author

Werner Schulte, Sep 14 2024

Keywords

Comments

Conjecture: This triangle seen as a sequence yields a permutation of the natural numbers.

Examples

			Triangle T(n, k) for 1 <= k <= n starts:
n \k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   3   2
   3 :   7   4   5
   4 :  11   6   9   8
   5 :  17  10  15  12  13
   6 :  23  14  21  16  19  18
   7 :  31  20  29  22  27  24  25
   8 :  39  26  37  28  35  30  33  32
   9 :  49  34  47  36  45  38  43  40  41
  10 :  59  42  57  44  55  46  53  48  51  50
  11 :  71  52  69  54  67  56  65  58  63  60  61
  12 :  83  62  81  64  79  66  77  68  75  70  73  72
  etc.

Crossrefs

Main diagonal is A000982.
Column 1 is A047838(n+1).
Column 2 is 2*A033638.
Cf. A246696 (permutation by row), A246697 (row sums), A376583 (parity).

Programs

  • PARI
    T(n,k)=(2*n^2+4*n+1-(-1)^n)/4-k-(1+(-1)^k)*(n-k)

Formula

T(n, k) = T(n, k-1) - (-1)^k * (2*n - 2*k + 1) for 2 <= k <= n.
T(n, k) = T(n, k-2) + 2 * (-1)^k for 3 <= k <= n.
Row sums: Sum_{k=1..n} T(n, k) = (n^3 + n) / 2 + (n - 1) * (1 - (-1)^n) / 4.
G.f.: x*y*(1 + x + x^2*(1 - y)^2 - 3*x^5*y^2 + 2*x^6*y^3 + x^4*y*(4 + y) - x^3*(1 + 4*y + y^2))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Sep 16 2024
From Ruud H.G. van Tol, Sep 22 2024: (Start)
T(n, 1) = A047838(n+1).
T(n, 2) = A033638(n) * 2.
T(n, n) = A000982(n) = (T(n, 1) + T(n, 2) - 1) / 2 for n >= 2. (End)
Previous Showing 21-22 of 22 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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