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, Length 2 paralleogram unique dam configuration.
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+3)*(4*n-2) - (4*n-2)*(4*n-1)/2 + 4 for n > 1. [corrected by Christian Krause, Feb 28 2026]
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)
E.g.f.: 3 + 5*x + exp(x)*(8*x^2 + 18*x - 3). - Stefano Spezia, Mar 01 2026
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 20, 49, 99, 165}, 50] (* Paolo Xausa, Jun 29 2026 *)
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
KEYWORD
nonn,easy
AUTHOR
Craig Knecht, Jun 15 2018
STATUS
approved