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 A173530 #12 Jun 02 2025 02:41:51 %S A173530 0,1,3,7,11,15,23,39,47,51,59,75,91,107,139,203,219,223,231,247,263, %T A173530 279,311,375,407,423,455,519,583,647,775,1031,1063,1067,1075,1091, %U A173530 1107,1123,1155,1219,1251,1267,1299,1363,1427,1491,1619 %N A173530 Number of ON cells after n generations of three-dimensional cellular automaton related to Sierpinski's triangle and the toothpick sequences (See Comments for definition). %C A173530 The structure is similar to Sierpinski's triangle but in this case we are in 3-D. %C A173530 The triangles of the new generation are arranged on planes that are orthogonal with respect to the planes of the previous generation. %C A173530 Rules: %C A173530 If n is odd then the triangles are arranged on planes that are parallel to the plane XZ. %C A173530 If n is even then the triangles are arranged on planes that are parallel to the plane YZ. %C A173530 The sequence A173531 (The first differences) gives the number of triangles added at the n-th stage. %C A173530 Example: %C A173530 We start with no triangles. %C A173530 At round 1 we place a triangle anywhere in the space on the plane XZ. %C A173530 At round 2 we place two other triangles on planes that are parallel to the plane YZ. %C A173530 At round 3 we place four other triangles on planes that are parallel to the plane XZ. %C A173530 And so on... %C A173530 It appears that the three-dimensional pattern has a recursive, fractal (or fractal-like) structure. An animation can show the fractal (or fractal-like) behavior. %C A173530 Note that the triangles can be replaced by V-toothpicks or L-toothpicks. More generally, the triangles can be replaced by any polytoothpick formed by two toothpicks connected by one of its vertices, with an angle greater than zero degrees and less than 180 degrees. %C A173530 In this structure every polytoothpick has two components, so after n stages the structure has 2 * a (n) components. %C A173530 Note that for n <= 11, in all cases (using triangles or polytoothpicks), one of the views of the 3-D structure is equal to the toothpick structure of A139250 (See illustrations). %C A173530 See the entries A139250, A161206 and A172310 for more information about the growth of toothpicks, V-toothpicks and L-toothpicks. %H A173530 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.] %H A173530 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a> %F A173530 Partial sums of A173531. %Y A173530 Cf. A047999, A139250, A161206, A172310, A173531, A173532. %K A173530 nonn %O A173530 0,3 %A A173530 _Omar E. Pol_, Oct 10 2010