A116513 Number of distinct hexagons of n points chosen from triangular lattice A_2 with sides parallel to the principal axes of that lattice. Degenerate sides (of length 1) are permitted.

1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 3, 5, 4, 5, 6, 5, 4, 7, 6, 6, 7, 6, 6, 9, 7, 7, 8, 8, 8, 10, 6, 8, 11, 10, 9, 12, 7, 10, 12, 10, 8, 13, 11, 12, 13, 10, 10, 15, 12, 13, 12, 12, 12, 18, 11, 13, 15, 12, 14, 18, 13, 14, 18

Offset: 1

Author

Allan C. Wechsler, Mar 23 2006

Keywords

• nonn

Comments

This sequence is to the lattice A2 as sequence A038548 is to the lattice D2; presumably other lattices have analogous sequences.

a(n) is also the number of 4-tuples (p,b,c,d) of nonnegative integers satisfying b <= c <= d, b + c + d < p, and n = t(p) - t(b) - t(c) - t(d) where t(x) is the x-th triangular number (A000217).

Examples

			a(7) = 3 because (reading the rows of the hexagons) 7 = 3+4 = 2+3+2.
		

Links

• Scott Reynolds, Table of n, a(n) for n = 1..250
• Nino Bašić, Patrick W. Fowler, Tomaž Pisanski, Stratified Enumeration of Convex Benzenoids, MATCH Commun. Math. Comput. Chem. 80 (2018) 153-172.

Crossrefs

Cf. A038548.

Extensions

More terms up to n=32 from Allan C. Wechsler, Mar 31 2006

Corrected and extended to n=36 by Allan C. Wechsler, Feb 15 2008

More terms up to n=68. and b-file to n=250 from Scott Reynolds, Mar 30 2012