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A286429
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Highest elevation of an island above sea level in a number square.
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1
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0, 0, 0, 0, 16, 27, 40, 55, 72, 91, 112, 135, 160, 187, 216, 247, 280, 315, 352, 391, 432, 475, 520, 567, 616, 667, 720, 775, 832, 891, 952, 1015, 1080, 1147, 1216, 1287, 1360, 1435, 1512, 1591, 1672, 1755, 1840, 1927, 2016, 2107, 2200, 2295, 2392
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OFFSET
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0,5
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COMMENTS
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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.
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.
A number square contains the numbers 1 to n^2 without repeats.
The larger terms in this sequence are a(n) = n*(n+6) or A028560.
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LINKS
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FORMULA
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G.f.: x^4*(16 - 21*x + 7*x^2) / (1 - x)^3.
a(n) = n^2 + 2*n - 8 for n>3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6.
(End)
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EXAMPLE
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For the 6 X 6 number square the largest value is 36 which is assigned to the single-cell island.
I only include the pertinent moat, spillway, and island values for the 6 X 6 example.
( 1 2 3 )
( 8 36 4 9 )
( 7 6 5 )
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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