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

1,1

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

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

Links

Table of n, a(n) for n=1..16.

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

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 A355148

Adjacent sequences: A275110 A275111 A275112 * A275114 A275115 A275116

Keyword

nonn,more

Author

Kival Ngaokrajang, Jul 17 2016

Extensions

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

Status

approved