%I
%S 12,14,15,16,16,17,18,18,19,19
%N a(n) is the minimal number of squares needed to enclose n squares with a wall so that there is a gap of at least one cell between the wall and the enclosed cells.
%C Inspired by beehive construction in which wax is used in the most efficient way. This problem is likened to construction of a fence around a house with minimum materials and maximum enclosed area. I conjectured that a specific house pattern shall be selected. See illustration in links.
%C If the conjecture in A261491 is true (i.e., A261491(n) is the number of squares required to enclose n squares without a gap), then a(n) = A261491(n) + 8.  _Charlie Neder_, Jul 11 2018
%H Kival Ngaokrajang, <a href="/A275113/a275113_1.pdf">Illustration of initial terms</a>
%e a(1) = 12:
%e ++++
%e  1 2 3
%e ++++++
%e 12  4
%e ++ ++ ++
%e 11  1  5
%e ++ ++ ++
%e 10  6
%e ++++++
%e  9 8 7
%e ++++
%e .
%e a(2) = 14:
%e +++++
%e  1 2 3 4
%e +++++++
%e 14  5
%e ++ +++ ++
%e 13  1 2  6
%e ++ +++ ++
%e 12  7
%e +++++++
%e 1110 9 8
%e +++++
%e .
%e a(3) = 15:
%e ++++
%e  1 2 3
%e ++++++
%e 15  4
%e ++ ++ +++
%e 14  3  5
%e ++ +++ ++
%e 13  1 2  6
%e ++ +++ ++
%e 12  7
%e +++++++
%e 1110 9 8
%e +++++
%Y Cf. A235382, A261491.
%K nonn,more
%O 1,1
%A _Kival Ngaokrajang_, Jul 17 2016
