OFFSET
1,2
COMMENTS
Construct a tetrahedron so rows have length j and the top left corner of each layer is directly underneath that of the previous layer (see Example section). Place a "1" in the top layer (P(1,1,1) = 1); in each successive layer starting in the top left corner (P(n,1,1)) and continuing horizontally until each successive row is complete: add the least positive integer so that no row, column or diagonal (in any horizontal or vertical direction) contains a repeated term. Here, the following definitions apply:
"row" means a horizontal line (read left to right) on a layer;
"horizontal column" means a line on a layer read vertically (downward) WITHIN a layer;
"vertical column" means a vertical line (read downward) ACROSS layers; and
"diagonal" means a diagonal line with slope 1 or -1 in any possible plane.
Conjecture: all infinite lines (i.e., all vertical columns and some multi-layer diagonals) are permutations of the natural numbers (while this has been proven for rows and columns in A269526, proofs here will require more subtle analysis).
EXAMPLE
Layers start P(1,1,1):
Layer 1: 1
----
Layer 2: 2
3 4
-------
Layer 3: 5
1 6
2 5 3
----------
Layer 4: 3
4 7
8 9 1
6 10 2 5
-------------
Layer 4, Row 3, Column 2 = P(4,3,2) = 9.
P(4,2,2) = 7 because all coefficients < 7 have appeared in at least one row, column or diagonal to P(4,2,2): P(3,2,1) = 1; P(3,3,1)= 2; P(3,3,3) and P(4,1,1) = 3; P(2,2,2) and P(4,2,1) = 4; P(3,1,1) and P(3,3,2) = 5; and P(3,2,2) = 6.
Expanding successive layers (read by rows):
1
2, 3, 4
5, 1, 6, 2, 5, 3
3, 4, 7, 8, 9, 1, 6, 10, 2, 5
6, 2, 5, 1, 3, 4, 4, 7, 6, 3, 5, 1, 8, 4, 2
4, 7, 8, 6, 10, 2, 8, 9, 5, 1, 10, 2, 11, 12, 5, 9, 3, 7, 13, 1, 6
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Bob Selcoe, Dec 12 2016
STATUS
approved