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.

12, 14, 15, 16, 16, 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22

Offset: 1

Author

Kival Ngaokrajang, Jul 17 2016

Keywords

• nonn
• more

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

The conjecture in A261491 holds through a(16). - David Consiglio, Jr., Nov 10 2022

Examples

			     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|
     +--+--+--+--+
		

Links

• David Consiglio, Jr., Python Program
• David Consiglio, Jr., Illustration of a(11)
• David Consiglio, Jr., Illustration of a(12)
• David Consiglio, Jr., Illustration of a(13)
• David Consiglio, Jr., Illustration of a(14)
• David Consiglio, Jr., Illustration of a(15)
• David Consiglio, Jr., Illustration of a(16)
• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A235382, A261491.

Extensions

a(11)-a(16) from David Consiglio, Jr., Nov 10 2022