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.

0, 2, 19, 76, 215, 481, 946, 1691, 2789, 4356, 6525, 9397, 13128, 17874, 23768, 31071, 39953, 50551, 63141, 77947, 95234, 115223, 138305, 164501, 194344, 228218, 266165, 308688, 356104, 408731, 467166, 531616, 602362, 679952, 764821, 857517

Offset

0,2

Comments

The chains are counted up to congruence.

Proof that a(n) = 3*A190313(n) + 2*A189978(n):

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).

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).

Links

Table of n, a(n) for n=0..35.

Alec Jones, Examples for n = 1 to 2

Formula

a(n) = 3*A190313(n) + 2*A189978(n).

Crossrefs

Cf. A190313, A189978.

Sequence in context: A219121 A054209 A256112 * A317274 A226019 A057326

Adjacent sequences: A272050 A272051 A272052 * A272054 A272055 A272056

Keyword

nonn

Author

Alec Jones, Apr 18 2016

Status

approved