

A286430


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


2



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 twolevel system will contain more water than a random 3level 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.


LINKS



FORMULA

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 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)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



