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 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. 1

%I

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

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

%C Each pair of stones has at most one group that counts going through them. _David A. Corneth_, Aug 01 2016

%C a(n) >= n and a(n+m) <= a(n)+a(m), i.e., 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 lim_{n-> oo} a(n)/n = 1. See arXiv link. - _Chai Wah Wu_, Aug 25 2016

%H Hong-Chang Wang, <a href="/A273916/a273916.png">Illustration of initial terms</a>

%H 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 From _M. F. Hasler_, Jul 30 2016: (Start)

%e One can get n=3 groups using a(3) = 9 stones (O) as follows:

%e O O O O The 3 groups are:

%e . O O . (1) the first line,

%e . O . . (2) the second column,

%e O O . . (3) the antidiagonal.

%e See the link for more examples. (End)

%K nonn,more,nice

%O 0,2

%A _Jiangshan Sun_, _Jason Y.S. Chiu_, _Hong-Chang Wang_, Jun 03 2016

%E Edited by _N. J. A. Sloane_, Jul 29 2016

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Last modified November 14 22:17 EST 2019. Contains 329134 sequences. (Running on oeis4.)