

A286430


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


1



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
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OFFSET

0,5


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


LINKS

Table of n, a(n) for n=0..32.
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


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(n1)  3*a(n2) + a(n3) for n>2.
(End)


EXAMPLE

For this 5X5 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 9*8  sum8 = 36 units retained.
( 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)


CROSSREFS

Cf. A054247, A201126, A268311.
Sequence in context: A255094 A261284 A111163 * A279901 A271629 A171675
Adjacent sequences: A286427 A286428 A286429 * A286431 A286432 A286433


KEYWORD

nonn


AUTHOR

Craig Knecht, May 09 2017


STATUS

approved



