OFFSET

1,2

COMMENTS

Construct a tetrahedron so rows have length n-j+1, 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 2

6 7

3

-----------

Layer 4: 3 4 5 6

2 8 3

1 5

7

-----------

Layer 4, Row 1, Column 3 = P(4,1,3) = 5.

P(4,1,4) = 6 because all coefficients < 6 have appeared in at least one row, column or diagonal to P(4,1,4): P(1,1,1) = 1; P(3,1,3)= 2; P(2,1,2) and P(4,1,1) = 3; P(4,1,2) = 4; and P(4,1,3) = 5.

Expanding successive layers (read by rows):

1

2, 3, 4

5, 1, 2, 6, 7, 3

3, 4, 5, 6, 2, 8, 3, 1, 5, 7

6, 7, 1, 4, 5, 9, 10, 2, 8, 4, 6, 7, 3, 2, 10

4, 5, 6, 3, 1, 7, 1, 3, 9, 10, 2, 7, 8, 11, 1, 11, 9, 4, 5, 6, 8

CROSSREFS

KEYWORD

nonn,tabf

AUTHOR

Bob Selcoe, Dec 12 2016

STATUS

approved