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
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
KEYWORD
nonn,more
AUTHOR
Kival Ngaokrajang, Jul 17 2016
EXTENSIONS
a(11)-a(16) from David Consiglio, Jr., Nov 10 2022
STATUS
approved