A351699 T(n,k) is the number of non-congruent maximal subsets of a grid of n X k lattice points (k <= n), such that no two points are at the same distance, and the points do not fit into a smaller grid. The size of the subsets is given by A351700. T(n,k) and A351700 are triangles read by rows.
1, 1, 1, 1, 2, 1, 1, 1, 5, 10, 1, 5, 28, 7, 21, 2, 19, 8, 104, 330, 2, 1, 4, 70, 15, 110, 574, 1, 3, 30, 272, 205, 4, 71, 563, 1991, 4, 68, 50, 1001, 113, 1130, 4, 76, 383, 9, 8, 362, 35, 1150, 23, 363, 3975, 7, 38, 8, 18, 1082, 415, 2, 638, 7503, 23, 515, 5802, 2, 2, 150, 62, 4238, 120, 1, 55, 1776, 17277, 26, 481, 2388
Offset: 1
Examples
The triangle begins:
#
# 1: 1 Counting grids n X k.
( 1 ) Two lines per side length n:
# 2: 2 2 1. for other side k = 1, 2, ...
( 1 1 ) maximal number of points
# 3: 2 3 3 2. number of configurations
( 1 2 1 )
# 4: 3 4 4 4 Example: 28 figures with
( 1 1 5 10 ) 4 points on 5 X 3
# 5: 3 4 4 5 5
( 1 5 28 7 21 )
# 6: 3 4 5 5 5 6
( 2 19 8 104 330 2 )
# 7: 4 5 5 6 6 6 7
( 1 4 70 15 110 574 1 )
# 8: 4 5 5 6 7 7 7 7
( 3 30 272 205 4 71 563 1991 )
# 9: 4 5 6 6 7 7 8 8 8
( 4 68 50 1001 113 1130 4 76 383 )
#10: 4 6 6 7 7 8 8 8 9 9
( 9 8 362 35 1150 23 363 3975 7 38 )
#11: 4 6 6 7 8 8 8 9 9 9 10
( 8 18 1082 415 2 638 7503 23 515 5802 2 )
#
# Grid n X k configurations with
# distinct distances
.
.
All T(6,3) = 8 configurations
0 1 2 3 4 5 0 1 2 3 4 5
------------------- -------------------
2 | . X X . X . 2 | . . . . X .
1 | . . . . . X 1 | . . . . . X
0 | X . . . . . 0 | X . X . . X
y /------------------- y /-------------------
x 0 1 2 3 4 5 x 0 1 2 3 4 5
{1,2,4,5,8,9,10,17,20,26} dist^2 {1,2,4,5,8,9,10,20,25,26}
0 1 2 3 4 5 0 1 2 3 4 5
------------------- -------------------
2 | . . X . X . 2 | . X . X . .
1 | . . . . . X 1 | X . . . . .
0 | X X . . . . 0 | X . . . . X
y /------------------- y /-------------------
x 0 1 2 3 4 5 x 0 1 2 3 4 5
{1,2,4,5,8,10,13,17,20,26} dist^2 {1,2,4,5,8,10,13,20,25,26}
0 1 2 3 4 5 0 1 2 3 4 5
------------------- -------------------
2 | . . . . X . 2 | . . X . X .
1 | X . . . . X 1 | X . . . . X
0 | X . X . . . 0 | X . . . . .
y /------------------- y /-------------------
x 0 1 2 3 4 5 x 0 1 2 3 4 5
{1,2,4,5,8,10,17,20,25,26} dist^2 {1,2,4,5,8,10,17,20,25,26}
0 1 2 3 4 5 0 1 2 3 4 5
------------------- -------------------
2 | . . X . . X 2 | X . . . . X
1 | . . . . . . 1 | . . . . . .
0 | X X . . . X 0 | X . . X X .
y /------------------- y /-------------------
x 0 1 2 3 4 5 x 0 1 2 3 4 5
{1,4,5,8,9,13,16,20,25,29} dist^2 {1,4,5,8,9,13,16,20,25,29}
.
Links
- Hugo Pfoertner, Table of n, a(n) for n = 1..91, rows 1..13 of triangle, flattened
Extensions
Completed row 8 and new rows 9-12 from Hugo Pfoertner, Jul 12 2022
Comments