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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.
5
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
1,5
COMMENTS
Configurations of points differing by any combination of rotation and reflection are counted only once.
LINKS
Hugo Pfoertner, Table of n, a(n) for n = 1..91, rows 1..13 of triangle, flattened
EXAMPLE
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}
.
KEYWORD
nonn,tabl,hard
AUTHOR
EXTENSIONS
Completed row 8 and new rows 9-12 from Hugo Pfoertner, Jul 12 2022
STATUS
approved