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%I #63 Nov 13 2022 12:29:07
%S 12,14,15,16,16,17,18,18,19,19,20,20,20,21,21,22
%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
%C The conjecture in A261491 holds through a(16). - _David Consiglio, Jr._, Nov 10 2022
%H David Consiglio, Jr., <a href="/A275113/a275113_1.py.txt">Python Program</a>
%H David Consiglio, Jr., <a href="/A275113/a275113.txt">Illustration of a(11)</a>
%H David Consiglio, Jr., <a href="/A275113/a275113_1.txt">Illustration of a(12)</a>
%H David Consiglio, Jr., <a href="/A275113/a275113_2.txt">Illustration of a(13)</a>
%H David Consiglio, Jr., <a href="/A275113/a275113_3.txt">Illustration of a(14)</a>
%H David Consiglio, Jr., <a href="/A275113/a275113_4.txt">Illustration of a(15)</a>
%H David Consiglio, Jr., <a href="/A275113/a275113_5.txt">Illustration of a(16)</a>
%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 |11|10| 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 |11|10| 9| 8|
%e +--+--+--+--+
%Y Cf. A235382, A261491.
%K nonn,more
%O 1,1
%A _Kival Ngaokrajang_, Jul 17 2016
%E a(11)-a(16) from _David Consiglio, Jr._, Nov 10 2022
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