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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A316323 The square array in A305615 read by antidiagonals.

Original entry on oeis.org

0, 1, 3, 4, 2, 8, 9, 7, 6, 15, 16, 14, 5, 13, 24, 25, 23, 12, 11, 22, 35, 36, 34, 21, 10, 20, 33, 48, 49, 47, 32, 19, 18, 31, 46, 63, 64, 62, 45, 30, 17, 29, 44, 61, 80, 81, 79, 60, 43, 28, 27, 42, 59, 78, 99, 100, 98, 77, 58, 41, 26, 40, 57, 76, 97, 120, 121, 119, 96, 75, 56, 39, 38, 55, 74, 95, 118, 143
Offset: 0

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Author

N. J. A. Sloane, Jul 03 2018

Keywords

Examples

			The array in A305615 begins:
  ^
  |
  4 |... ... ... ... ...
    +---------------+
  3 | 9  14  12  10 |...
    +-----------+   |
  2 | 4   7   5 |11 |...
    +-------+   |   |
  1 | 1   2 | 6 |13 |...
    +---+   |   |   |
  0 | 0 | 3 | 8 |15 |...
    +---+---+---+---+---
      0   1   2   3   4 ...
The first few antidiagonals are:L
0,
1, 3,
4, 2, 8,
9, 7, 6, 15,
16, 14, 5, 13, 24,
25, 23, 12, 11, 22, 35,
36, 34, 21, 10, 20, 33, 48,
...

Crossrefs

Cf. A305615, A269501, A269780.

Formula

If 1 is added to every term we get the array in A269780, which has an explicit formula for the (i,j)-th term.

A269501 Subsequence immediately following the instances of n in the sequence is n, n-1, ..., 1, n+1, n+2, ....

Original entry on oeis.org

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

Views

Author

Franklin T. Adams-Watters, Mar 04 2016

Keywords

Comments

The sequence includes every ordered pair of positive integers exactly once as consecutive terms of the sequence. Through n = k^2, it has every pair i,j with 0 < i,j <= k.
Can be regarded as an irregular triangle where row k contains 1, k, k, k-1, k, k-2, ..., 2, k, with 2n-1 terms.
See A305615 for an essentially identical sequence: a(n) = A305615(n)+1. - N. J. A. Sloane, Jul 03 2018

Examples

			The first 3 occurs as a(5), so a(6) = 3, the first term of 3, 2, 1, 4, 5, 6, .... The second 3 is thus a(6), so a(7) = 2. The third 3 is a(8), so a(9) = 1. The fourth 3 is a(12), now we start incrementing, and a(13) = 4.
The triangle starts:
  1
  1, 2, 2
  1, 3, 3, 2, 3
  1, 4, 4, 3, 4, 2, 4
  1, 5, 5, 4, 5, 3, 5, 2, 5

Crossrefs

Cf. A003059, A060747 (row lengths), A000326 (row sums), A097291, A269780.
See also A315615.

Programs

  • PARI
    a(n) = my(r = if(n<=0, 0, sqrtint(n-1)+1));if((n-r)%2,r,(r^2-n)/2 + 1)

Formula

Let r = ceiling(sqrt(n)) = A003059(n). If n and r have the same parity, a(n) = (r^2-n)/2 + 1; otherwise a(n) = r.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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