OFFSET
0,3
COMMENTS
The water retention model for mathematical surfaces led to definitions for a lake and a pond. These lakes and ponds divide the square up in interesting ways. This sequence looks at the spillway heights in zero or one bend minimal area lakes.
A lake has dimensions of (n-2) X (n-2) when the square is n X n. All other water retaining areas are ponds.
A number square contains the numbers 1 to n^2 without repeats.
The larger terms are a(n)= n^2+6 or A114949.
LINKS
Craig Knecht, Minimal lake types in a 7x7 square.
Craig Knecht, Minimal lake area in a square
Wikipedia, Water retention on mathematical surfaces
FORMULA
Conjectures from Colin Barker, May 07 2017: (Start)
G.f.: x^2*(6 - 8*x + 3*x^2 + x^3) / (1 - x)^3.
a(n) = 7 - 2*n + n^2 for n>2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>5.
(End)
EXAMPLE
For the 4 X 4 square a example of a smallest lake is shown. The values 1,2,3 form the lake. The pathway of least resistance off the square is the spillway value 10.
( 4 16 15 5)
(10 1 2 14)
( 6 11 3 13)
( 7 8 12 9)
CROSSREFS
KEYWORD
nonn
AUTHOR
Craig Knecht, May 04 2017
STATUS
approved