A279478 A 3-dimensional variant of A269526 "Infinite Sudoku": expansion (read first by layer, then by row) of "Type 2" tetrahedron P(n,j,k). (See A269526 and Comments section below for definition.)

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, 8, 2, 11, 5, 6, 3, 4, 10, 12, 4, 7, 9, 5, 2, 13, 14, 8, 12, 1, 3, 7, 9, 12, 19, 1, 4, 11, 6

Offset: 1

Author

Bob Selcoe, Dec 12 2016

Keywords

• nonn
• tabf

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).

Examples

			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

Cf. A269526.

Cf. A279049, A279477 ("Type 1" tetrahedron).

Cf. A000217 (triangular numbers).