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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
12, 14, 15, 16, 16, 17, 18, 18, 19, 19 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

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

LINKS

Table of n, a(n) for n=1..10.

Kival Ngaokrajang, Illustration of initial terms

EXAMPLE

     a(1) = 12:

     +--+--+--+

     | 1| 2| 3|

  +--+--+--+--+--+

  |12|        | 4|

  +--+  +--+  +--+

  |11|  | 1|  | 5|

  +--+  +--+  +--+

  |10|        | 6|

  +--+--+--+--+--+

     | 9| 8| 7|

     +--+--+--+

.

      a(2) = 14:

     +--+--+--+--+

     | 1| 2| 3| 4|

  +--+--+--+--+--+--+

  |14|           | 5|

  +--+  +--+--+  +--+

  |13|  | 1| 2|  | 6|

  +--+  +--+--+  +--+

  |12|           | 7|

  +--+--+--+--+--+--+

     |11|10| 9| 8|

     +--+--+--+--+

.

     a(3) = 15:

     +--+--+--+

     | 1| 2| 3|

  +--+--+--+--+--+

  |15|        | 4|

  +--+  +--+  +--+--+

  |14|  | 3|     | 5|

  +--+  +--+--+  +--+

  |13|  | 1| 2|  | 6|

  +--+  +--+--+  +--+

  |12|           | 7|

  +--+--+--+--+--+--+

     |11|10| 9| 8|

     +--+--+--+--+

CROSSREFS

Cf. A235382, A261491.

Sequence in context: A075477 A075482 A248019 * A043651 A043701 A290001

Adjacent sequences:  A275110 A275111 A275112 * A275114 A275115 A275116

KEYWORD

nonn,more

AUTHOR

Kival Ngaokrajang, Jul 17 2016

STATUS

approved

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Last modified January 26 01:48 EST 2020. Contains 331270 sequences. (Running on oeis4.)