A395035 a(n) is the number of different ways for two particles to reach a corner of a triangular grid of side n units, starting simultaneously from the other two corners 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 before arriving the destination.

1, 2, 0, 28, 166, 1244, 7556, 50838, 330392, 2217834, 14959408, 102346420, 707121818, 4934693936, 34738105738, 246529508304, 1762353205164, 12683158904472, 91836400707618, 668719858612542, 4894628181505660, 35997340001521430, 265914520027198698

Offset

1,2

Comments

Both particles move at 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 from A to B gives a unique Q-path from B to A and R-path from C to A by transforming a grid point (h,k) on P-path into; (n-h-k,k) on Q-path and (k,n-h-k) on R-path. A Q-path and a R-path do not meet if r-th coordinate of Q is different from r-th coordinate of R. The r-th and (r+1)-th coordinates of Q must not appear swapped in R 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 (Q,R) of strings of coordinates

Example

n = 2 has 2 different ways of moving from the corners B,C to the corner A:
Grid paths from B to A:
  Q1:{(2,0), (1,1), (0,1), (0,0)},
  Q2:{(2,0), (1,0), (0,1), (0,0)}.
Grid paths from C to A:
  R1:{(0,2), (0,1), (1,0), (0,0)},
  R2:{(0,2), (1,1), (1,0), (0,0)}.
Set of valid pairs of grid paths:
  {(Q1,R1), (Q2,R2)}
Therefore a(2) = 2.

Crossrefs

Cf. A394098, A394698.

Sequence in context: A356564 A007218 A092177 * A345762 A340534 A104466

Adjacent sequences: A395032 A395033 A395034 * A395036 A395037 A395038

Keyword

nonn,walk

Author

Janaka Rodrigo, Apr 10 2026

Extensions

More terms from Sean A. Irvine, Apr 19 2026

Status

approved