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