A286430 Least volume of water to surround the largest possible island in a number square.

0, 0, 0, 0, 36, 78, 136, 210, 300, 406, 528, 666, 820, 990, 1176, 1378, 1596, 1830, 2080, 2346, 2628, 2926, 3240, 3570, 3916, 4278, 4656, 5050, 5460, 5886, 6328, 6786, 7260

Offset: 0

Author

Craig Knecht, May 09 2017

Keywords

• nonn

Comments

The water retention model for mathematical surfaces showed that a random two-level system will contain more water than a random 3-level system when the size of the square is > 52 X 52. It has also been the subject of Zimmermann's programming contest in 2010 and a Wikipedia page as noted below. The number square is a simple environment in which to explore the interaction of volumes, heights, and areas of lakes, ponds, islands, and spillways in the square.

A number square contains the numbers for 1 to n^2 without repeats in an n X n square.

This sequence is 4*A000217 for a(n) > 8.

Examples

			For this 5 X 5 square the numbers 1 to 25 are used without repeats. The values 1 through 8 form the moat.  The spillway value is 9.  The volume of water retained is 36 units.
( 24 23 22 21 20)
( 18  1  2  3 19)
( 17  8 25  4  9)
( 16  7  6  5 15)
( 14 13 12 11 10)
		

Links

• Craig Knecht, 3D graphic.
• Craig Knecht, Least volume of water to surround the largest island in a number square.
• Craig Knecht, Number range for each cell type.
• Wikipedia, Water retention on mathematical surfaces

Crossrefs

Cf. A054247, A201126, A268311.

Formula

Conjectures from Colin Barker, Jan 20 2018: (Start)

G.f.: 2*x^4*(18 - 15*x + 5*x^2) / (1 - x)^3.

a(n) = 28 - 30*n + 8*n^2 for n>3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.

(End)