A308178 Scan an infinite 45-degree triangular chessboard (cells (x,y) with 0 <= y <= x) by upwards antidiagonals, filling in each cell with the smallest nonnegative number already placed that cannot be seen by a chess queen at (x,y); sequence gives numbers along the successive antidiagonals.

0, 1, 2, 3, 3, 0, 4, 1, 5, 5, 2, 4, 6, 7, 0, 2, 7, 4, 1, 3, 8, 5, 2, 4, 1, 9, 6, 3, 0, 2, 10, 11, 7, 1, 3, 6, 11, 8, 10, 9, 4, 5, 12, 9, 6, 8, 0, 10, 7, 13, 10, 12, 5, 6, 11, 8, 14, 15, 8, 7, 9, 3, 5, 11, 15, 12, 9, 6, 8, 0, 13, 10, 16, 13, 11, 12, 5, 1, 14, 7

Offset

0,3

Comments

The 0's occur in positions (x,y) = (2k,k), k >= 0.

Column y=1 is A263313; the main diagonal is A308180.

After 13 steps, the y=2 column appears to become quasi-periodic with a saltus of 4. That is, the first differences appear to become periodic with period (-1, -2, 1, 6).

There is a very similar triangle in A274650.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10200 (antidiagonals for x+y = 0..200)

Rémy Sigrist, Colored representation of the first 1000 antidiagonals (black pixels correspond to zeros)

Rémy Sigrist, PARI program for A308178

Example

Start of chessboard showing antidiagonals 0 through 12:
y =  0, 1, 2, 3, 4, 5, 6, 7, ...
--------------------------------
x=0  0,
x=1  1, 3,
x=2  2, 0, 5,
x=3  3, 1, 4, 2,
x=4  4, 2, 0, 3, 1,
x=5  5, 7, 1, 4, 2, 6,
x=6  6, 4, 2, 0, 3, 5, 7,
x=7  7, 5, 3, 1, 4, 10, ...,
x=8  8, 6, 7, 9, 0, ...,
x=9  9, 11, 10, 8, ...,
x=10 10, 8, 6, ...,
x=11 11, 9, ...,
x=12 12, ...,
x=13 ...,
The first few antidiagonals are:
0,
1,
2, 3,
3, 0,
4, 1, 5,
5, 2, 4,
6, 7, 0, 2,
7, 4, 1, 3,
8, 5, 2, 4, 1,
9, 6, 3, 0, 2,
...

Prog

(PARI) See Links section.

Crossrefs

Reading the triangle across rows gives A308179.

Cf. A263313, A274650, A308180

Sequence in context: A332915 A192498 A360665 * A127572 A021815 A238525

Adjacent sequences: A308175 A308176 A308177 * A308179 A308180 A308181

Keyword

nonn,tabf

Author

N. J. A. Sloane, May 28 2019

Extensions

More terms from Rémy Sigrist, May 29 2019

Status

approved