

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.


1



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
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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 2segment polygonal chain (XY,YZ). By the sideangleside criterion for triangle congruence, the triangle to which XY and YZ belong is determined up to congruence, and so the proposed formula does not overcount. 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



