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Least volume of water to surround the largest possible island in a number square.
2

%I #28 Sep 14 2021 04:06:22

%S 0,0,0,0,36,78,136,210,300,406,528,666,820,990,1176,1378,1596,1830,

%T 2080,2346,2628,2926,3240,3570,3916,4278,4656,5050,5460,5886,6328,

%U 6786,7260

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

%C 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.

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

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

%H Craig Knecht, <a href="/A286430/a286430_1.png">3D graphic</a>.

%H Craig Knecht, <a href="/A286430/a286430.png">Least volume of water to surround the largest island in a number square</a>.

%H Craig Knecht, <a href="/A286430/a286430_2.png">Number range for each cell type</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Water_retention_on_mathematical_surfaces">Water retention on mathematical surfaces</a>

%F Conjectures from _Colin Barker_, Jan 20 2018: (Start)

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

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

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

%F (End)

%e 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.

%e ( 24 23 22 21 20)

%e ( 18 1 2 3 19)

%e ( 17 8 25 4 9)

%e ( 16 7 6 5 15)

%e ( 14 13 12 11 10)

%Y Cf. A054247, A201126, A268311.

%K nonn

%O 0,5

%A _Craig Knecht_, May 09 2017