A279049 A 3-dimensional variant of A269526 "Infinite Sudoku": expansion (read first by layer, then by row) of square pyramid P(n,j,k). (See A269526 and "Comments" below for definition).

1, 2, 3, 4, 5, 6, 1, 2, 7, 8, 4, 3, 2, 7, 3, 5, 6, 7, 9, 10, 3, 1, 1, 6, 5, 2, 5, 4, 1, 6, 4, 7, 8, 5, 6, 2, 11, 9, 10, 3, 12, 8, 13, 4, 5, 3, 2, 10, 7, 1, 8, 6, 11, 3, 2, 8, 10, 1, 4, 7, 5, 6, 3, 2, 11, 9, 8, 4, 1, 12, 8, 13, 6, 7, 5, 14, 9, 11, 3, 1, 4, 15, 5, 6, 7, 2, 7, 8, 1, 10, 4

Offset: 1

Author

Bob Selcoe, Dec 04 2016

Keywords

• nonn
• tabf

Comments

Comments: Construct a square pyramid so the top left corners of each layer are directly underneath each other. 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

			Example:
Layers start P(1,1,1):
Layer 1:          1
                  -----
Layer 2:          2  3
                  4  5
                  --------
Layer 3:          6  1  2
                  7  8  4
                  3  2  7
                  -----------
Layer 4:          3  5  6  7
                  9 10  3  1
                  1  6  5  2
                  5  4  1  6
                  -----------
Layer 4, Row 2, Column 1 = P(4,2,1) = 9.
P(4,3,3) = 5 because all coefficients < 5 have appeared in at least one row, column or diagonal to P(4,3,3): P(4,2,4) and P(4,3,1) = 1; P(2,1,1) and P(3,3,2) = 2; P(4,1,1) and P(4,2,3) = 3; and P(3,2,3) = 4.
Expanding successive layers (read by rows):
1
2, 3, 4, 5
6, 1, 2, 7, 8, 4, 3, 2, 7
3, 5, 6, 7, 9, 10, 3, 1, 1, 6, 5, 2, 5, 4, 1, 6
4, 7, 8, 5, 6, 2, 11, 9, 10, 3, 12, 8, 13, 4, 5, 3, 2, 10, 7, 1, 8, 6, 11, 3, 2
		

Crossrefs

Cf. A269526.

Cf. A000330 (square pyramidal numbers).