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 A170876 #16 Feb 24 2021 02:48:19 %S A170876 0,1,5,21,37,53,117,197,261,405,565,789,965,1221,1541,1941,2453,2933, %T A170876 3621,4389,5093,5909,6805,7925,9093,10629,12197,14133,15733,17717, %U A170876 19493,21605,23909,26453,29109,32117,35013,38085,41285 %N A170876 Number of toothpicks after n stages of 3-D toothpick structure defined in Comments. %C A170876 We are in 3-D, and we are placing ordinary toothpicks, as in A139250. %C A170876 We start with one toothpick in the z direction %C A170876 We place toothpicks at any free end, as in A139250. %C A170876 We always place new toothpicks in pairs, two perpendicular toothpicks that are perpendicular to the original toothpick %C A170876 The toothpicks are always in 2 out of the 3 (x, y or z) directions. %C A170876 The initial values are as follows (this should be checked!): %C A170876 n:.0..1..2..3..4..5..6 %C A170876 ---------------------------- %C A170876 x..0..0..2..4..8..4.24 (Number added in x direction) %C A170876 y..0..0..2..4..8..4.24 (Number added in y direction) %C A170876 z..0..1..0..8..0..8.16 (Number added in z direction) %C A170876 ---------------------------- %C A170876 ...0..1..4.16.16.16.64 (Total number added at n-th stage, A170876) %C A170876 ---------------------------- %C A170876 a..0..1..5.21.37.53.117 (Total so far, this sequence) %C A170876 ---------------------------- %H A170876 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 A170876 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> %H A170876 R. J. Mathar, <a href="/A170876/a170876.cc">C++ program</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_2.eps">View after stage 1</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_3.eps">View after stage 2</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_4.eps">View after stage 3</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_5.eps">View after stage 4</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_6.eps">View after stage 5</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_7.eps">View after stage 6</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_8.eps">View after stage 7</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_9.eps">View after stage 8</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_10.eps">View after stage 9</a> %H A170876 R. J. Mathar, <a href="/A170876/a170876_11.eps">View after stage 10</a> %H A170876 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp053.jpg">Illustration of initial terms</a> %e A170876 At stage 2 we have a horizontal cross, a vertical toothpick then another horizontal cross, for a total of 5 toothpicks. %e A170876 Then we add 8 vertical toothpicks at the ends of the crosses and 8 horizontal toothpicks in the same planes as the crosses, for a total of 21 toothpicks. %Y A170876 Cf. A139250, A170875 (first differences), A160160, A160170. For another version see A170837. %K A170876 nonn %O A170876 0,3 %A A170876 _N. J. A. Sloane_, Jan 05 2010, based on email from _R. J. Mathar_, Jun 02 2009 Revised by _R. J. Mathar_, Jan 08 2010, Jan 09 2010