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 A275113 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. 1

%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 |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

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Last modified February 26 18:29 EST 2020. Contains 332293 sequences. (Running on oeis4.)