A394698 a(n) is the number of different ways to move from A to B along grid paths in an equilateral triangular grid ABC of side n units such that the distance between A and each successive point of the path increases while the direction of movement changes at each grid point.

1, 2, 5, 10, 24, 56, 135, 332, 826, 2084, 5310, 13652, 35372, 92262, 242081, 638514, 1692036, 4502674, 12027492, 32238194, 86681150, 233733662, 631915388, 1712568790, 4651671690, 12661092950, 34527973186, 94330427526, 258143998494, 707547095008, 1942180671839

Offset

1,2

Comments

Let ABC be an equilateral triangular grid of side n containing n^2 unit equilateral triangles. The x axis is defined along AB and the 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 should not have same x coordinates or same y coordinates or same value for x + y.

Links

Sean A. Irvine, Table of n, a(n) for n = 1..500

Janaka Rodrigo, Python code for sets of strings of coordinates

Example

n = 3 has 5 distinct sets of strings of coordinates:
  {(0,0), (1,0), (1,1), (2,0), (3,0)},
  {(0,0), (0,1), (1,1), (2,0), (2,1), (3,0)},
  {(0,0), (1,0), (1,1), (2,1), (3,0)},
  {(0,0), (1,0), (1,1), (2,0), (2,1), (3,0)},
  {(0,0), (0,1), (1,1), (2,0), (3,0)}.
Therefore a(3) = 5.

Crossrefs

Cf. A393796, A394097, A394098.

Sequence in context: A359068 A151514 A321007 * A253013 A192471 A001431

Adjacent sequences: A394695 A394696 A394697 * A394699 A394700 A394701

Keyword

nonn,walk

Author

Janaka Rodrigo, Mar 29 2026

Status

approved