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A279478 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.)

%I A279478 #9 Dec 21 2016 11:15:23
%S A279478 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 A279478 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 A279478 4,7,9,5,2,13,14,8,12,1,3,7,9,12,19,1,4,11,6
%N A279478 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 A279478 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 A279478 "row" means a horizontal line (read left to right) on a layer;
%C A279478 "horizontal column" means a line on a layer read vertically (downward) WITHIN a layer;
%C A279478 "vertical column" means a vertical line (read downward) ACROSS layers; and
%C A279478 "diagonal" means a diagonal line with slope 1 or -1 in any possible plane.
%C A279478 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 A279478 Layers start P(1,1,1):
%e A279478 Layer 1:          1
%e A279478                   -----
%e A279478 Layer 2:          2  3
%e A279478                   4
%e A279478                   --------
%e A279478 Layer 3:          5  1  2
%e A279478                   6  7
%e A279478                   3
%e A279478                   -----------
%e A279478 Layer 4:          3  4  5  6
%e A279478                   2  8  3
%e A279478                   1  5
%e A279478                   7
%e A279478                   -----------
%e A279478 Layer 4, Row 1, Column 3 = P(4,1,3) = 5.
%e A279478 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 A279478 Expanding successive layers (read by rows):
%e A279478 1
%e A279478 2, 3, 4
%e A279478 5, 1, 2, 6, 7, 3
%e A279478 3, 4, 5, 6, 2, 8, 3,  1, 5, 7
%e A279478 6, 7, 1, 4, 5, 9, 10, 2, 8, 4,  6, 7, 3, 2, 10
%e A279478 4, 5, 6, 3, 1, 7, 1,  3, 9, 10, 2, 7, 8, 11, 1, 11, 9, 4, 5, 6, 8
%Y A279478 Cf. A269526.
%Y A279478 Cf. A279049, A279477 ("Type 1" tetrahedron).
%Y A279478 Cf. A000217 (triangular numbers).
%K A279478 nonn,tabf
%O A279478 1,2
%A A279478 _Bob Selcoe_, Dec 12 2016