A273916 The Bingo-4 problem: minimal number of stones that must be placed on an infinite square grid to produce n groups of exactly 4 stones each. Groups consist of adjacent stones in a horizontal, vertical or diagonal line.

0, 4, 7, 9, 11, 12, 12, 14, 15, 16, 16, 18, 19, 20, 22, 24

Offset: 0

Author

Jiangshan Sun, Jason Y.S. Chiu, Hong-Chang Wang, Jun 03 2016

Keywords

• nonn
• more
• nice

Comments

You are permitted to put 5 or more adjacent stones in a line, but cannot count them as a group.

Each pair of stones has at most one group that counts going through them. - David A. Corneth, Aug 01 2016

a(n) >= n and a(n+m) <= a(n) + a(m), e.g., a(16) <= a(10) + a(6) = 28. Placing stones in a 4 X k rectangular array shows that a(3k) <= 4(k+2). Fekete's subadditive lemma shows that 1 <= lim_{n->oo} a(n)/n <= 4/3 exists. - Chai Wah Wu, Jul 31 2016

Limit_{n->oo} a(n)/n = 1. See arXiv link. - Chai Wah Wu, Aug 25 2016

Examples

			From _M. F. Hasler_, Jul 30 2016: (Start)
One can get n=3 groups using a(3) = 9 stones (O) as follows:
   O O O O     The 3 groups are:
   . O O .     (1) the first line,
   . O . .     (2) the second column,
   O O . .     (3) the antidiagonal.
See the link for more examples. (End)
		

Links

• Hong-Chang Wang, Illustration of initial terms.
• Chai Wah Wu, Minimal number of points on a grid forming patterns of blocks, arXiv:1608.07247 [math.CO], 2016.

Crossrefs

See also the 4-trees-in-a-row orchard problem, A006065.

Extensions

Edited by N. J. A. Sloane, Jul 29 2016