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

%I A286430 #28 Sep 14 2021 04:06:22
%S A286430 0,0,0,0,36,78,136,210,300,406,528,666,820,990,1176,1378,1596,1830,
%T A286430 2080,2346,2628,2926,3240,3570,3916,4278,4656,5050,5460,5886,6328,
%U A286430 6786,7260
%N A286430 Least volume of water to surround the largest possible island in a number square.
%C A286430 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 A286430 A number square contains the numbers for 1 to n^2 without repeats in an n X n square.
%C A286430 This sequence is 4*A000217 for a(n) > 8.
%H A286430 Craig Knecht, <a href="/A286430/a286430_1.png">3D graphic</a>.
%H A286430 Craig Knecht, <a href="/A286430/a286430.png">Least volume of water to surround the largest island in a number square</a>.
%H A286430 Craig Knecht, <a href="/A286430/a286430_2.png">Number range for each cell type</a>.
%H A286430 Wikipedia, <a href="http://en.wikipedia.org/wiki/Water_retention_on_mathematical_surfaces">Water retention on mathematical surfaces</a>
%F A286430 Conjectures from _Colin Barker_, Jan 20 2018: (Start)
%F A286430 G.f.: 2*x^4*(18 - 15*x + 5*x^2) / (1 - x)^3.
%F A286430 a(n) = 28 - 30*n + 8*n^2 for n>3.
%F A286430 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
%F A286430 (End)
%e A286430 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 A286430 ( 24 23 22 21 20)
%e A286430 ( 18  1  2  3 19)
%e A286430 ( 17  8 25  4  9)
%e A286430 ( 16  7  6  5 15)
%e A286430 ( 14 13 12 11 10)
%Y A286430 Cf. A054247, A201126, A268311.
%K A286430 nonn
%O A286430 0,5
%A A286430 _Craig Knecht_, May 09 2017