A395370 a(n) is the number of pairs (B->A, C->A) of non-intersecting paths in a triangular grid ABC of side n units, such that the distance from the starting point to each point of a path is always increasing.

1, 1, 5, 39, 332, 4337, 65355, 1084401, 20253143, 407155395, 8669715855

Offset

1,3

Comments

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).

Points on the path satisfy x + y <= n, y <= n/2 and x^2 + y^2 + x*y should increase when moving to the next grid point.

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 pair of grid paths does not contain common points other than at starting and finishing.

Links

Table of n, a(n) for n=1..11.

Pontus von Brömssen, Illustration for n=4.

Pontus von Brömssen, Illustration for n=5.

Janaka Rodrigo, Python code for pairs (Q,R) of strings of coordinates

Example

n = 2 has only one way of reaching a corner from other two corners:
Grid paths from B to A:
 Q1:{(2,0), (1,1), (0,1), (0,0)},
 Q2:{(2,0), (1,0), (0,1), (0,0)},
 Q3:{(2,0), (1,0), (0,0)}.
Grid paths from C to A:
 R1:{(0,2), (1,1), (1,0), (0,0)},
 R2:{(0,2), (0,1), (1,0), (0,0)},
 R3:{(0,2), (0,1), (0,0)}.
Set of valid pairs of grid paths:
 {(Q3,R3)}.
Therefore a(2) = 1.

Crossrefs

Cf. A394698, A394098, A395035, A395369, A395371.

Sequence in context: A356622 A273019 A390408 * A244039 A387929 A328554

Adjacent sequences: A395367 A395368 A395369 * A395371 A395372 A395373

Keyword

nonn,walk,more

Author

Janaka Rodrigo, Apr 20 2026

Extensions

a(9)-a(10) from Sean A. Irvine, May 14 2026

a(11) from Pontus von Brömssen, May 15 2026

Status

approved