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%I #9 Dec 21 2016 11:15:23
%S 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,
%T 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,
%U 4,7,9,5,2,13,14,8,12,1,3,7,9,12,19,1,4,11,6
%N 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.)
%C 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:
%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 Layers start P(1,1,1):
%e Layer 1: 1
%e -----
%e Layer 2: 2 3
%e 4
%e --------
%e Layer 3: 5 1 2
%e 6 7
%e 3
%e -----------
%e Layer 4: 3 4 5 6
%e 2 8 3
%e 1 5
%e 7
%e -----------
%e Layer 4, Row 1, Column 3 = P(4,1,3) = 5.
%e 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.
%e Expanding successive layers (read by rows):
%e 1
%e 2, 3, 4
%e 5, 1, 2, 6, 7, 3
%e 3, 4, 5, 6, 2, 8, 3, 1, 5, 7
%e 6, 7, 1, 4, 5, 9, 10, 2, 8, 4, 6, 7, 3, 2, 10
%e 4, 5, 6, 3, 1, 7, 1, 3, 9, 10, 2, 7, 8, 11, 1, 11, 9, 4, 5, 6, 8
%Y Cf. A269526.
%Y Cf. A279049, A279477 ("Type 1" tetrahedron).
%Y Cf. A000217 (triangular numbers).
%K nonn,tabf
%O 1,2
%A _Bob Selcoe_, Dec 12 2016
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