A296339 On an infinite 60-degree sector of hexagonal graph paper, fill in cells by antidiagonals so that each contains the least nonnegative integer such that no line of edge-adjacent cells contains a repeated term.

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

Offset: 0

Author

N. J. A. Sloane, Dec 10 2017

Keywords

• nonn
• tabl

Comments

To find the number to enter in a cell (assuming the sector is oriented as in the illustration in the link), look at all the numbers in the cells directly above the cell, in the cells to the "North-West", and in the cells to the "South-West", and take their "mex" (the smallest missing number).

The 0-cells in the array all lie on a perfectly straight line (in contrast to the situation in A274528). Also a(n) = 0 iff n = 2*m*(m+1) for some m.

Examples

			The initial rows are as follows (however, this does not show the adjancies between the cells correctly - for that, see the illustration in the link):
   0;
   1,  2;
   2,  0,  1;
   3,  1,  2,  4;
   4,  5,  0,  3,  6;
   5,  3,  4,  6,  7,  8;
   6,  4,  5,  0,  3,  9,  7;
   7,  8,  3,  1,  2,  4,  5,  9;
   8,  6,  7,  2,  0,  1,  9,  4,  3;
   9,  7,  8,  5,  1,  2,  6, 10, 11, 12;
  10, 11,  6,  9,  4,  0, ...
  ...
For example, referring to the illustration in the link and NOT to the triangle here, consider the first 5 in the array. The reason this is 5 is because in the column of cells above that cell we can see 2,0,1, to the NW we see 3, and to the SW we see 4, and the smallest missing number is 5.
		

Links

• Rémy Sigrist, Rows n = 0..200, flattened
• F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
• Rémy Sigrist, Colored representation of the first 2^9 rows of the triangle
• Rémy Sigrist, PARI program for A296339
• N. J. A. Sloane, Illustration of initial rows of the sector.

Crossrefs

Two analogs of this for an infinite square chessboard are A269526 (which uses positive numbers) and A274528 (which uses nonnegative numbers).

For the right edge see A296340.

The second column is A004483. - Rémy Sigrist, Dec 11 2017

The third and fourth columns are A004482 and A298801.

See also A274820.

Programs

Mathematica

ab = Table[0, {13}];
nw = ab;
A296339 = Reap[For[s = 1, s <= Length[ab], s++, sw = 0; For[c = 1, c <= s, c++, x = BitOr[ab[[c]], BitOr[nw[[s-c+1]], sw]]; v = IntegerExponent[x+1, 2]; Sow[v]; p = 2^v; sw += p; ab[[c]] += p; nw[[s-c+1]] += p]]][[2, 1]] (* Jean-François Alcover, Dec 18 2017, after Rémy Sigrist *) (* I changed the first line, which was ab = Table[0, 13];, to make this compatible with older versions of MMA - N. J. A. Sloane, Feb 03 2018 *)

PARI

See Links section.

Extensions

More terms from Rémy Sigrist, Dec 11 2017