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A285766 Maximum spillway height for a zero or one bend minimal area lake in a number square. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 26 21:07 EST 2021. Contains 349344 sequences. (Running on oeis4.)