This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A169786 #11 Oct 03 2013 23:16:32
%S A169786 1,1,2,1,1,2,3,5,3,3,1,1,2,3,5,5,5,9,13,7,3,5,8,6,4,1,1,2,3,5,5,5,9,
%T A169786 13,9,5,9,15,19,17,21,29,15,3,5,8,10,10,14,22,18,8,8,13,10,5,1,1,2,3,
%U A169786 5,5,5,9,13,9,5,9,15,19,17,21,29,17,5,9,15,19,19,27,43,43,25,21,37,51,47,51,63
%N A169786 Triangle read by rows: T(n,k) is number of cells that turn from OFF to ON at stage k of the growth of the obtuse triangle of hexagons described in the comment.
%C A169786 Consider the tiling of the plane by hexagons, where each cell has 6 neighbors, as in the A151723, A151724, A170905, A170906.
%C A169786 Assume the hexagons are oriented so that each one has a pair of vertical edges.
%C A169786 We label the cells in the usual way by Eisenstein integers, complex numbers r+sw, where r,s in Z, w = exp(2 pi i / 3) (see Conway and Sloane, pp. 52-53).
%C A169786 Initially all cells are OFF.
%C A169786 For x >= 1, define a roughly triangular region B_x by declaring the cells {sw: s >= 1}, {r-w: r >= -1}, {x-1-i+iw: 0 <= i <= x-2}, {x-1-i+(i+1)w: 0 <= i <= x-3} to be permanently OFF.
%C A169786 In other words, B_x consists of 0 plus the cells {r+sw: 0 <= s <= x-3, 1 <= r <= x-s-2}.
%C A169786 At stage 1, the "corner" cell 0 is turned ON; thereafter, a cell in B_x is turned ON if it has exactly one ON neighbor. Once a cell is ON it stays ON.
%C A169786 T(n,k) is the number of cells in B_{2^n} that are turned from OFF to ON at stage k (1 <= k <= 2^n-1).
%C A169786 Row n has 2^n-1 terms.
%D A169786 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd ed., 1988, see pp. 52-53.
%e A169786 Example: B_8:
%e A169786 .W W W
%e A169786 ..W 6 W W
%e A169786 ...W 5 5 W W
%e A169786 ....W 4 X 4 W W
%e A169786 .....W 3 3 4 X W W
%e A169786 ......W 2 X 4 X 6 W W
%e A169786 .......1 2 3 4 5 6 7 W
%e A169786 ........W W W W W W W
%e A169786 W = permanently OFF, X = OFF, ON cells are labeled with the stage at which they turned ON.
%e A169786 Triangle begins:
%e A169786 1,
%e A169786 1, 2, 1,
%e A169786 1, 2, 3, 5, 3, 3, 1,
%e A169786 1, 2, 3, 5, 5, 5, 9, 13, 7, 3, 5, 8, 6, 4, 1,
%e A169786 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 15, 3, 5, 8, 10, 10, 14, 22, 18, 8, 8, 13, 10, 5, 1,
%e A169786 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 5, 9, 15, 19, 19, 27, 43, 43, 25, 21, 37, 51, 47, 51, 63, 31, 3, 5, 8, 10, 10, 14, 22, 22, 14, 14, 24, 34, 36, 38, 50, 42, 16, 8, 13, 18, 20, 24, 36, 36, 20, 15, 21, 15, 6, 1,
%e A169786 1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 5, 9, 15, 19, 19, 27, 43, 43, 25, 21, 37, 51, 47, 51, 63, 33, 5, 9, 15, 19, 19, 27, 43, 43, 27, 27, 47, 67, 71, 75, 99, 91, 41, 21, 37, 51, 55, 71, 111, 127, 87, 59, 87, 125, 119, 119, 133, 63, 3, 5, 8, 10, 10, 14, 22, 22, 14, 14, 24, 34, 36, 38, 50, 46, 22, 14, 24, 34, 38, 46, 70, 86, 68, 46, 58, 88, 98, 98, 114, 92, 32, 8, 13, 18, 20, 24, 36, 44, 36, 28, 38, 58, 70, 74, 88, 88, 52, 23, 21, 31, 38, 44, 60, 64, 44, 30, 33, 21, 7, 1,
%e A169786 ...
%e A169786 The rows converge to A169787.
%K A169786 nonn,tabf
%O A169786 1,3
%A A169786 _David Applegate_ and _N. J. A. Sloane_, May 12 2010