A394098 - OEIS

A394098

a(n) is the number of different ways for two particles to reach a corner of an equilateral triangular grid of side n units, starting simultaneously from the other two corners and moving at the same constant speed along the grid lines, 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, 7, 19, 567, 6752, 130323, 2662965, 57111881, 1298864135, 31289028259, 769777090611, 19615121770513, 511835666156710, 13622327637676078, 369086594750646175, 10159222725719863129, 283326524903029194315, 7996211072500624733903, 228098194638401870363735

(list; graph; refs; listen; history; text; internal format)

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.

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..19.

Janaka Rodrigo, Python code for sets of pairs (Q.R)

EXAMPLE

n = 2 has 7 different ways of moving from the corners B, C to the corner A:

Grid paths from B to A:

Q1:{(2,0), (1,0), (0,0)},

Q2:{(2,0), (1,1), (0,1), (0,0)},

Q3:{(2,0), (1,0), (0,1), (0,0)}.

Grid paths from C to A:

R1:{(0,2), (0,1), (0,0)},