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

%I A303295 #41 Jun 28 2019 03:18:14
%S A303295 0,20,49,99,165,247,345,459,589,735,897,1075,1269,1479,1705,1947,2205,
%T A303295 2479,2769,3075,3397,3735,4089,4459,4845,5247,5665,6099,6549,7015,
%U A303295 7497,7995,8509,9039,9585,10147,10725,11319,11929,12555
%N A303295 a(n) is the maximum water retention of a height-3 length-n number parallelogram with maximum water area.
%C A303295 A number parallelogram contains numbers from 1 to the triangular area of the parallelogram without duplicate numbers.
%C A303295 This sequence applies the water retention model for mathematical surfaces to the triangular grid.
%C A303295 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.
%C A303295 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.
%C A303295 Five magic polyiamond tilings for a single numeric solution are noted in the link section.
%H A303295 Colin Barker, <a href="/A303295/b303295.txt">Table of n, a(n) for n = 0..1000</a>
%H A303295 Craig Knecht, <a href="/A303295/a303295_1.png">Example for the sequence.</a>
%H A303295 Craig Knecht, <a href="/A303295/a303295_2.png">Five magic polyiamond tilings of a single numeric solution.</a>
%H A303295 Craig Knecht, <a href="/A303295/a303295_3.png">Length 2 paralleogram unique dam configuration.</a>
%H A303295 Craig Knecht, <a href="/A303295/a303295.png">Magic polyiamond tiling H3 L4 Parallelogram with 99 units retained.</a>
%H A303295 Craig Knecht, <a href="/A303295/a303295_5.png">Water retention using a pentagonal tile.</a>
%H A303295 Wikipedia, <a href="http://en.wikipedia.org/wiki/Water retention on mathematical surfaces">Water retention on mathematical surfaces</a>
%H A303295 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A303295 a(n) = ((4*n+7)*(4*n+2)) - (4*n+2) * (4*n+3)/2 + 4 for n > 2.
%F A303295 From _Colin Barker_, Jun 15 2018: (Start)
%F A303295 G.f.: x*(20 - 11*x + 12*x^2 - 5*x^3) / (1 - x)^3.
%F A303295 a(n) = -3 + 10*n + 8*n^2 for n>1.
%F A303295 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
%F A303295 (End)
%o A303295 (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
%Y A303295 Cf. A261347.
%K A303295 nonn,easy
%O A303295 0,2
%A A303295 _Craig Knecht_, Jun 15 2018