|
|
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
|
|
|
0, 4, 7, 9, 11, 12, 12, 14, 15, 16, 16, 18, 19, 20, 22, 24
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
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
|
|
LINKS
|
|
|
EXAMPLE
|
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 4-trees-in-a-row orchard problem, A006065.
|
|
KEYWORD
|
nonn,more,nice
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|