cp's OEIS Frontend

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.

%I A273916 #54 Sep 19 2023 08:20:02
%S A273916 0,4,7,9,11,12,12,14,15,16,16,18,19,20,22,24
%N 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.
%C A273916 You are permitted to put 5 or more adjacent stones in a line, but cannot count them as a group.
%C A273916 Each pair of stones has at most one group that counts going through them. - _David A. Corneth_, Aug 01 2016
%C A273916 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
%C A273916 Limit_{n->oo} a(n)/n = 1. See arXiv link. - _Chai Wah Wu_, Aug 25 2016
%H A273916 Hong-Chang Wang, <a href="/A273916/a273916.png">Illustration of initial terms</a>.
%H A273916 Chai Wah Wu, <a href="http://arxiv.org/abs/1608.07247">Minimal number of points on a grid forming patterns of blocks</a>, arXiv:1608.07247 [math.CO], 2016.
%e A273916 From _M. F. Hasler_, Jul 30 2016: (Start)
%e A273916 One can get n=3 groups using a(3) = 9 stones (O) as follows:
%e A273916    O O O O     The 3 groups are:
%e A273916    . O O .     (1) the first line,
%e A273916    . O . .     (2) the second column,
%e A273916    O O . .     (3) the antidiagonal.
%e A273916 See the link for more examples. (End)
%Y A273916 See also the 4-trees-in-a-row orchard problem, A006065.
%K A273916 nonn,more,nice
%O A273916 0,2
%A A273916 _Jiangshan Sun_, _Jason Y.S. Chiu_, _Hong-Chang Wang_, Jun 03 2016
%E A273916 Edited by _N. J. A. Sloane_, Jul 29 2016