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

0,2

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).

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)

Prog

(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

Crossrefs

Cf. A261347.

Sequence in context: A264444 A053245 A115882 * A277553 A260093 A304832

Adjacent sequences: A303292 A303293 A303294 * A303296 A303297 A303298

Keyword

nonn,easy

Author

Craig Knecht, Jun 15 2018

Status

approved