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 A303295 a(n) is the maximum water retention of a height-3 length-n number parallelogram with maximum water area. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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. 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

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Last modified November 16 02:33 EST 2018. Contains 317252 sequences. (Running on oeis4.)