A395034 a(n) is the number of different ways for two particles to reach each other's starting corner on a triangular grid of side n units while changing direction at each grid point, such that the distance between each particle and its own starting corner is always increasing, and the two particles never meet.

0, 2, 0, 32, 274, 1574, 6856, 61618, 334358, 2654388, 15055592, 114358536, 732505262, 5383308430, 36017638220, 262291328996, 1832624867022, 13327293899442, 95234646907040, 696070597359534, 5063050890337060, 37253184779069542, 274174108861808900

Offset

1,2

Comments

Both particles start at the same time and move with the same constant speed along the grid lines of the triangular grid.

Let ABC be an equilateral triangular grid of side n containing n^2 unit equilateral triangles. X axis is defined along AB and Y axis is defined along AC such that starting point A has coordinates (0,0), finishing point B has coordinates (n,0) and C is at (0,n).

A grid point (x,y) on a path of the moving point P satisfies the following conditions when generating string of coordinates of grid points on a valid path.

x + y <= n, y <= n/2 and x^2 + y^2 + x*y should increase when moving to the next grid point. Any three consecutive grid points of a string cannot have same x coordinates or same y coordinates or same value for x + y.

Each of P-path gives a unique Q-path by transforming a grid point (h, k) on P-path into (n - h - k, k) on Q-path. A P-path and a Q-path do not meet if r-th coordinate of P is different from r-th coordinate of Q. The r-th and (r+1)-th coordinates of P must not appear swapped in Q at positions r and r+1. Paths do not need to have equal length.

Links

Table of n, a(n) for n=1..23.

Janaka Rodrigo, Python Code for pairs of grid paths

Janaka Rodrigo, Python Code for pairs (P,Q) of strings of coordinates

Example

n = 2 has 2 different ways of reaching to other's starting corner:
Grid paths from A to B:
  P1:{(0,0), (0,1), (1,1), (2,0)},
  P2:{(0,0), (1,0), (1,1), (2,0)}.
Grid paths from B to A:
  Q1:{(2,0), (1,0), (0,1), (0,0)},
  Q2:{(2,0), (1,1), (0,1), (0,0)}.
Set of valid pairs of grid paths:
  {(P1,Q1), (P2,Q2)}
Therefore a(2) = 2.

Crossrefs

Cf. A394097, A394698.

Sequence in context: A156452 A156473 A156504 * A340545 A009665 A053552

Adjacent sequences: A395031 A395032 A395033 * A395035 A395036 A395037

Keyword

nonn,walk

Author

Janaka Rodrigo, Apr 10 2026

Extensions

More terms from Sean A. Irvine, Apr 19 2026

Status

approved