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A285766
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Maximum spillway height for a zero or one bend minimal area lake in a number square.
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2
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0, 0, 6, 10, 15, 22, 31, 42, 55, 70, 87, 106, 127, 150, 175, 202, 231, 262, 295, 330, 367, 406, 447, 490, 535, 582, 631, 682, 735, 790, 847, 906, 967, 1030, 1095, 1162, 1231, 1302, 1375, 1450, 1527, 1606, 1687, 1770, 1855, 1942, 2031, 2122, 2215, 2310, 2407
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OFFSET
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0,3
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COMMENTS
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The water retention model for mathematical surfaces led to definitions for a lake and a pond. These lakes and ponds divide the square up in interesting ways. This sequence looks at the spillway heights in zero or one bend minimal area lakes.
A lake has dimensions of (n-2) X (n-2) when the square is n X n. All other water retaining areas are ponds.
A number square contains the numbers 1 to n^2 without repeats.
The larger terms are a(n)= n^2+6 or A114949.
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LINKS
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FORMULA
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G.f.: x^2*(6 - 8*x + 3*x^2 + x^3) / (1 - x)^3.
a(n) = 7 - 2*n + n^2 for n>2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>5.
(End)
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EXAMPLE
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For the 4 X 4 square a example of a smallest lake is shown. The values 1,2,3 form the lake. The pathway of least resistance off the square is the spillway value 10.
( 4 16 15 5)
(10 1 2 14)
( 6 11 3 13)
( 7 8 12 9)
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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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