A379777 Array A(n, k), n, k >= 0, read by upward antidiagonals; for any v >= 0, the value appears twice in the array: in row A002262(v) and in row A002024(v+1); values in each row are given in strictly increasing order.

0, 0, 1, 1, 2, 3, 3, 2, 4, 6, 6, 4, 5, 7, 10, 10, 7, 5, 8, 11, 15, 15, 11, 8, 9, 12, 16, 21, 21, 16, 12, 9, 13, 17, 22, 28, 28, 22, 17, 13, 14, 18, 23, 29, 36, 36, 29, 23, 18, 14, 19, 24, 30, 37, 45, 45, 37, 30, 24, 19, 20, 25, 31, 38, 46, 55, 55, 46, 38, 31, 25, 20, 26, 32, 39, 47, 56, 66

Offset: 0

Author

Rémy Sigrist, Jan 02 2025

Keywords

• nonn
• tabl
• easy

Comments

This sequence was inspired by the game Dobble: this game is based on cards with symbols such that two distinct cards always have exactly one common symbol. Here, two distinct rows have exactly one common term.

This square array combines two symetrical copies of the triangular view of A001477 (the nonnegative integers):

0 1 3 6 .

2 4 7 . 0 1 3 6 .

5 8 . 0 2 4 7 .

0 9 . -> 1 2 5 8 .

1 2 . 3 4 5 9 .

3 4 5 6 7 8 9 .

6 7 8 9 . . . . .

. . . . .

Examples

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9
  ----+---------------------------------------
    0 |  0   1   3   6  10  15  21  28  36  45
    1 |  0   2   4   7  11  16  22  29  37  46
    2 |  1   2   5   8  12  17  23  30  38  47
    3 |  3   4   5   9  13  18  24  31  39  48
    4 |  6   7   8   9  14  19  25  32  40  49
    5 | 10  11  12  13  14  20  26  33  41  50
    6 | 15  16  17  18  19  20  27  34  42  51
    7 | 21  22  23  24  25  26  27  35  43  52
    8 | 28  29  30  31  32  33  34  35  44  53
    9 | 36  37  38  39  40  41  42  43  44  54
   10 | 45  46  47  48  49  50  51  52  53  54
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10010
• Wikipedia, Dobble

Crossrefs

Cf. A000096, A000217, A001477, A002024, A002262.

Programs

PARI

A(n, k) = { my (x, y); if (n > k, x = n-1; y = k, x = k; y = n;); x*(x+1)/2 + y }

Formula

A(0, k) = A000217(k).

A(n, k) = A(k+1, n) = A000217(k) + n for any n in 0..k.

A(n, n) = A000096(n).