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

0,5

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.

Links

Table of n, a(n) for n=0..48.

Craig Knecht, Maximum island elevation above sea level in a number square

Wikipedia, Water retention on mathematical surfaces

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)

Example

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    )

Crossrefs

Cf. A054247, A201126.

Sequence in context: A088247 A366962 A032610 * A329206 A280935 A067650

Adjacent sequences: A286426 A286427 A286428 * A286430 A286431 A286432

Keyword

nonn

Author

Craig Knecht, May 09 2017

Status

approved