A303295 a(n) is the maximum water retention of a height-3 length-n number parallelogram with maximum water area.

0, 20, 49, 99, 165, 247, 345, 459, 589, 735, 897, 1075, 1269, 1479, 1705, 1947, 2205, 2479, 2769, 3075, 3397, 3735, 4089, 4459, 4845, 5247, 5665, 6099, 6549, 7015, 7497, 7995, 8509, 9039, 9585, 10147, 10725, 11319, 11929, 12555

Offset: 0

Author

Craig Knecht, Jun 15 2018

Keywords

• nonn
• easy

Comments

A number parallelogram contains numbers from 1 to the triangular area of the parallelogram without duplicate numbers.

This sequence applies the water retention model for mathematical surfaces to the triangular grid.

Magic polyiamond tiling is the tiling of a number shape with a single order of polyiamond. The sum of numbers in each polyiamond subspace is equal.

The height-three length-four parallelogram has an area of 24 unit triangles. The sum of the numbers from 1 to 24 is 300. Both 24 and 300 are divisible by four and six making magic polyiamond tilings possible with order four and six polyiamonds.

Five magic polyiamond tilings for a single numeric solution are noted in the link section.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Craig Knecht, Example for the sequence.
• Craig Knecht, Five magic polyiamond tilings of a single numeric solution.
• Craig Knecht, Length 2 paralleogram unique dam configuration.
• Craig Knecht, Magic polyiamond tiling H3 L4 Parallelogram with 99 units retained.
• Craig Knecht, Water retention using a pentagonal tile.
• Wikipedia, Water retention on mathematical surfaces
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A261347.

Programs

PARI

concat(0, Vec(x*(20 - 11*x + 12*x^2 - 5*x^3) / (1 - x)^3 + O(x^50))) \\ Colin Barker, Jun 15 2018

Formula

a(n) = ((4*n+7)*(4*n+2)) - (4*n+2) * (4*n+3)/2 + 4 for n > 2.

From Colin Barker, Jun 15 2018: (Start)

G.f.: x*(20 - 11*x + 12*x^2 - 5*x^3) / (1 - x)^3.

a(n) = -3 + 10*n + 8*n^2 for n>1.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.

(End)