OFFSET
2,1
COMMENTS
At most one king can be placed on each square.
Every third term is conjectured to be A014105. Other terms are A001105. A093353 is conjectured to be this sequence with repeated terms removed.
The above conjectures are true (see Beveridge link). - Colin Beveridge, Jan 13 2025
LINKS
Rob Pratt, Table of n, a(n) for n = 2..100
Colin Beveridge, Proof of the case where n is a multiple of 3
Matthew Scroggs, December 23
Matthew Scroggs, Friendly squares
Matthew Scroggs, Python code to compute A379726
Puzzling StackExchange, Minimum Number of Squares to Color
Dominic McCarty, Illustration of a(n) for n = 2..100
FORMULA
If n is not a multiple of 3, a(n) = 2*floor((n+2)/3)^2.
If n is a multiple of 3, it is conjectured that a(n)=2*(n/3)^2+n/3.
The above conjectures are true (see Beveridge link). - Colin Beveridge, Jan 13 2025
EXAMPLE
For a 3 by 3 chessboard, the three kings could be placed like this (where o is an empty square and k is a king):
ooo
kkk
ooo
For a 4 by 4 chessboard, the kings could be placed like this:
oooo
kkkk
okko
okko
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthew Scroggs, Dec 31 2024
EXTENSIONS
a(15)-a(100) via integer linear programming by Rob Pratt, Jan 02 2025
STATUS
approved
