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A286429 Highest elevation of an island above sea level in a number square.

%I A286429 #18 Jun 04 2017 02:42:30
%S A286429 0,0,0,0,16,27,40,55,72,91,112,135,160,187,216,247,280,315,352,391,
%T A286429 432,475,520,567,616,667,720,775,832,891,952,1015,1080,1147,1216,1287,
%U A286429 1360,1435,1512,1591,1672,1755,1840,1927,2016,2107,2200,2295,2392
%N A286429 Highest elevation of an island above sea level in a number square.
%C A286429 The water retention model for mathematical surfaces has previously looked at lakes and ponds. This sequence looks at the maximum possible height of an island above water level in a number square.
%C A286429 The smallest possible water elevation will always be composed of an eight-cell lake or pond with a spillway value of nine. This moat is not centered in a(n) > 5 but has the square's edge as one of its borders.
%C A286429 A number square contains the numbers 1 to n^2 without repeats.
%C A286429 The larger terms in this sequence are a(n) = n*(n+6) or A028560.
%H A286429 Craig Knecht, <a href="/A286429/a286429_2.png">Maximum island elevation above sea level in a number square</a>
%H A286429 Wikipedia, <a href="http://en.wikipedia.org/wiki/Water_retention_on_mathematical_surfaces">Water retention on mathematical surfaces</a>
%F A286429 Conjectures from _Colin Barker_, May 09 2017: (Start)
%F A286429 G.f.: x^4*(16 - 21*x + 7*x^2) / (1 - x)^3.
%F A286429 a(n) = n^2 + 2*n - 8 for n>3.
%F A286429 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6.
%F A286429 (End)
%e A286429 For the 6 X 6 number square the largest value is 36 which is assigned to the single-cell island.
%e A286429 I only include the pertinent moat, spillway, and island values for the 6 X 6 example.
%e A286429 ( 1   2   3    )
%e A286429 ( 8  36   4  9 )
%e A286429 ( 7   6   5    )
%Y A286429 Cf. A054247, A201126.
%K A286429 nonn
%O A286429 0,5
%A A286429 _Craig Knecht_, May 09 2017