

A273916


The Bingo4 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



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

0,2


COMMENTS

You are permitted to put 5 or more adjacent stones in a line, but cannot count it 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), 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
lim_{n> oo} a(n)/n = 1. See arXiv link.  Chai Wah Wu, Aug 25 2016


LINKS

Table of n, a(n) for n=0..15.
HongChang 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


EXAMPLE

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)


CROSSREFS

See also the 4treesinarow orchard problem, A006065.
Sequence in context: A272015 A310941 A310942 * A053169 A007656 A159619
Adjacent sequences: A273913 A273914 A273915 * A273917 A273918 A273919


KEYWORD

nonn,more,nice


AUTHOR

Jiangshan Sun, Jason Y.S. Chiu, HongChang Wang, Jun 03 2016


EXTENSIONS

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


STATUS

approved



