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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).

%I A279049 #11 Dec 08 2016 10:30:25
%S A279049 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,
%T A279049 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,
%U A279049 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
%N 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).
%C A279049 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:
%C A279049 "row" means a horizontal line (read left to right) on a layer;
%C A279049 "horizontal column" means a line on a layer read vertically (downward) WITHIN a layer;
%C A279049 "vertical column" means a vertical line (read downward) ACROSS layers; and
%C A279049 "diagonal" means a diagonal line with slope 1 or -1 in any possible plane.
%C A279049 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).
%e A279049 Example:
%e A279049 Layers start P(1,1,1):
%e A279049 Layer 1:          1
%e A279049                   -----
%e A279049 Layer 2:          2  3
%e A279049                   4  5
%e A279049                   --------
%e A279049 Layer 3:          6  1  2
%e A279049                   7  8  4
%e A279049                   3  2  7
%e A279049                   -----------
%e A279049 Layer 4:          3  5  6  7
%e A279049                   9 10  3  1
%e A279049                   1  6  5  2
%e A279049                   5  4  1  6
%e A279049                   -----------
%e A279049 Layer 4, Row 2, Column 1 = P(4,2,1) = 9.
%e A279049 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.
%e A279049 Expanding successive layers (read by rows):
%e A279049 1
%e A279049 2, 3, 4, 5
%e A279049 6, 1, 2, 7, 8, 4, 3, 2, 7
%e A279049 3, 5, 6, 7, 9, 10, 3, 1, 1, 6, 5, 2, 5, 4, 1, 6
%e A279049 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
%Y A279049 Cf. A269526.
%Y A279049 Cf. A000330 (square pyramidal numbers).
%K A279049 nonn,tabf
%O A279049 1,2
%A A279049 _Bob Selcoe_, Dec 04 2016