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A279049
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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).
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2
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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
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Layer 4: 3 5 6 7
9 10 3 1
1 6 5 2
5 4 1 6
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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
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CROSSREFS
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Cf. A000330 (square pyramidal numbers).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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