A298267 a(n) is the maximum number of heptiamonds in a hexagon of order n.

0, 3, 7, 13, 21, 30, 42, 54, 69, 85, 103, 123, 144, 168, 192, 219, 247, 277, 309, 342, 378, 414, 453, 493, 535, 579, 624, 672, 720, 771, 823, 877, 933, 990, 1050, 1110, 1173, 1237, 1303, 1371, 1440, 1512, 1584, 1659, 1735, 1813, 1893, 1974, 2058, 2142

Offset

0,2

Comments

There are 24 heptiamonds.

It would be nice if this idea could be generalized to state that the hexagon can contain the maximum number of polyiamonds of any given size.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Craig Knecht, H2 Hexagon with 3 heptiamonds packed in.

Craig Knecht, H3 hexagon with 7 heptiamonds packed in.

Craig Knecht, H4 H5 H6 H7 heptiamond packing.

Craig Knecht, Peripheral Buildouts.

Craig Knecht, Proof notes.

Formula

a(n) = floor((6*n^2)/7).

Conjectures from Colin Barker, Jan 20 2018: (Start)

G.f.: x*(1 + x)*(3 - 2*x + 4*x^2 - 2*x^3 + 3*x^4) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).

a(n) = 2*a(n-1) - a(n-2) + a(n-7) - 2*a(n-8) + a(n-9) for n>8.

(End)

Mathematica

Array[Floor[(6 #^2)/7] &, 50] (* Michael De Vlieger, Jan 20 2018 *)

Crossrefs

Cf. A033581 (The number of triangles in a hexagon), A291582 (hexiamond tiling).

Sequence in context: A077853 A256588 A025721 * A235532 A195020 A169627

Adjacent sequences: A298264 A298265 A298266 * A298268 A298269 A298270

Keyword

nonn

Author

Craig Knecht, Jan 15 2018

Status

approved