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A272053 a(n) is the number of equivalence classes of simple, open polygonal chains consisting of two segments and with all three vertices on the lattice points of an n X n grid.

%I A272053 #21 May 18 2016 11:17:45
%S A272053 0,2,19,76,215,481,946,1691,2789,4356,6525,9397,13128,17874,23768,
%T A272053 31071,39953,50551,63141,77947,95234,115223,138305,164501,194344,
%U A272053 228218,266165,308688,356104,408731,467166,531616,602362,679952,764821,857517
%N A272053 a(n) is the number of equivalence classes of simple, open polygonal chains consisting of two segments and with all three vertices on the lattice points of an n X n grid.
%C A272053 The chains are counted up to congruence.
%C A272053 Proof that a(n) = 3*A190313(n) + 2*A189978(n):
%C A272053 Let ABC be a lattice triangle in an n X n grid. If ABC is scalene, then the pairs (BA,AC), (AB,BC), and (AC, CB) form three inequivalent polygonal chains; likewise, if ABC is isosceles and AB is the base of the triangle, then (BA,AC) and (AC,CB) form two distinct polygonal chains, while (BC,CA) is congruent to (AB,BC).
%C A272053 Now consider an arbitrary 2-segment polygonal chain (XY,YZ). By the side-angle-side criterion for triangle congruence, the triangle to which XY and YZ belong is determined up to congruence, and so the proposed formula does not over-count. Thus a(n) = 3*A190313(n) + 2*A189978(n).
%H A272053 Alec Jones, <a href="/A272053/a272053_1.jpg">Examples for n = 1 to 2</a>
%F A272053 a(n) = 3*A190313(n) + 2*A189978(n).
%Y A272053 Cf. A190313, A189978.
%K A272053 nonn
%O A272053 0,2
%A A272053 _Alec Jones_, Apr 18 2016