A285766 Maximum spillway height for a zero or one bend minimal area lake in a number square.

0, 0, 6, 10, 15, 22, 31, 42, 55, 70, 87, 106, 127, 150, 175, 202, 231, 262, 295, 330, 367, 406, 447, 490, 535, 582, 631, 682, 735, 790, 847, 906, 967, 1030, 1095, 1162, 1231, 1302, 1375, 1450, 1527, 1606, 1687, 1770, 1855, 1942, 2031, 2122, 2215, 2310, 2407

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

Table of n, a(n) for n=0..50.

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

Cf. A054247, A201126, A268311.

Sequence in context: A315284 A022949 A049694 * A157344 A332765 A093773

Adjacent sequences: A285763 A285764 A285765 * A285767 A285768 A285769

Keyword

nonn

Author

Craig Knecht, May 04 2017

Status

approved