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A279477 A 3-dimensional variant of A269526 "Infinite Sudoku": expansion (read first by layer, then by row) of "Type 1" tetrahedron P(n,j,k). (See A269526 and Comments section below for definition.)

%I A279477 #8 Dec 21 2016 11:15:13
%S A279477 1,2,3,4,5,1,6,2,5,3,3,4,7,8,9,1,6,10,2,5,6,2,5,1,3,4,4,7,6,3,5,1,8,4,
%T A279477 2,4,7,8,6,10,2,8,9,5,1,10,2,11,12,5,9,3,7,13,1,6,9,5,1,3,11,7,1,4,3,
%U A279477 8,6,12,10,2,3,7,5,8,6,4,10,2,6,1,3,5,7,11
%N A279477 A 3-dimensional variant of A269526 "Infinite Sudoku": expansion (read first by layer, then by row) of "Type 1" tetrahedron P(n,j,k). (See A269526 and Comments section below for definition.)
%C A279477 Construct a tetrahedron so rows have length j 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 A279477 "row" means a horizontal line (read left to right) on a layer;
%C A279477 "horizontal column" means a line on a layer read vertically (downward) WITHIN a layer;
%C A279477 "vertical column" means a vertical line (read downward) ACROSS layers; and
%C A279477 "diagonal" means a diagonal line with slope 1 or -1 in any possible plane.
%C A279477 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 A279477 Layers start P(1,1,1):
%e A279477 Layer 1:          1
%e A279477                   ----
%e A279477 Layer 2:          2
%e A279477                   3  4
%e A279477                   -------
%e A279477 Layer 3:          5
%e A279477                   1  6
%e A279477                   2  5  3
%e A279477                   ----------
%e A279477 Layer 4:          3
%e A279477                   4  7
%e A279477                   8  9  1
%e A279477                   6 10  2  5
%e A279477                   -------------
%e A279477 Layer 4, Row 3, Column 2 = P(4,3,2) = 9.
%e A279477 P(4,2,2) = 7 because all coefficients < 7 have appeared in at least one row, column or diagonal to P(4,2,2): P(3,2,1) = 1; P(3,3,1)= 2; P(3,3,3) and P(4,1,1) = 3; P(2,2,2) and P(4,2,1) = 4; P(3,1,1) and P(3,3,2) = 5; and P(3,2,2) = 6.
%e A279477 Expanding successive layers (read by rows):
%e A279477 1
%e A279477 2, 3, 4
%e A279477 5, 1, 6, 2, 5, 3
%e A279477 3, 4, 7, 8, 9, 1, 6, 10, 2, 5
%e A279477 6, 2, 5, 1, 3, 4, 4, 7,  6, 3, 5, 1, 8, 4, 2
%e A279477 4, 7, 8, 6, 10, 2, 8, 9, 5, 1, 10, 2, 11, 12, 5, 9, 3, 7, 13, 1, 6
%Y A279477 Cf. A269526.
%Y A279477 Cf. A279049, A279478 ("Type 2" tetrahedron).
%Y A279477 Cf. A000217 (triangular numbers).
%K A279477 nonn,tabf
%O A279477 1,2
%A A279477 _Bob Selcoe_, Dec 12 2016