%I
%S 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,
%T 10,9,12,7,10,12,10,8,13,11,12,13,10,10,15,12,13,12,12,12,18,11,13,15,
%U 12,14,18,13,14,18
%N 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.
%C This sequence is to the lattice A2 as sequence A038548 is to the lattice D2; presumably other lattices have analogous sequences.
%C a(n) is also the number of 4tuples (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 xth triangular number (A000217).
%H Scott Reynolds, <a href="/A116513/b116513.txt">Table of n, a(n) for n = 1..250</a>
%H Nino Bašić, Patrick W. Fowler, Tomaž Pisanski, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match80/n1/match80n1_153172.pdf">Stratified Enumeration of Convex Benzenoids</a>, MATCH Commun. Math. Comput. Chem. 80 (2018) 153172.
%e a(7) = 3 because (reading the rows of the hexagons) 7 = 3+4 = 2+3+2.
%Y Cf. A038548.
%K nonn
%O 1,3
%A _Allan C. Wechsler_, Mar 23 2006
%E More terms up to n=32 from _Allan C. Wechsler_, Mar 31 2006
%E Corrected and extended to n=36 by _Allan C. Wechsler_, Feb 15 2008
%E More terms up to n=68. and bfile to n=250 from _Scott Reynolds_, Mar 30 2012
