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

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

Offset: 0

Author

Craig Knecht, May 09 2017

Keywords

• nonn

Comments

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.

Examples

			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    )
		

Links

• Craig Knecht, Maximum island elevation above sea level in a number square
• Wikipedia, Water retention on mathematical surfaces

Crossrefs

Cf. A054247, A201126.

Formula

Conjectures from Colin Barker, May 09 2017: (Start)

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)