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 A152998 #28 Feb 24 2021 02:48:18 %S A152998 0,1,3,5,7,11,17,21,23,27,33,39,47,61,77,85,87,91,97,103,111,125,141, %T A152998 151,159,173,191,211,241,285,325,341,343,347,353,359,367,381,397,407, %U A152998 415,429,447,467,497,541,581,599,607,621,639 %N A152998 Toothpick sequence on the semi-infinite square grid. %C A152998 Contribution from Omar E. Pol, Oct 01 2011 (Start): %C A152998 On the semi-infinite square grid, at stage 0, we start from a vertical half toothpick at [(0,0),(0,1)]. This half toothpick represents one of the two components of the first toothpick placed in the toothpick structure of A139250. Consider only the toothpicks of length 2, so a(0) = 0. %C A152998 At stage 1, we place an orthogonal toothpick of length 2 centered at the end, so a(1) = 1. %C A152998 In each subsequent stage, for every exposed toothpick end, place an orthogonal toothpick centered at that end. %C A152998 The sequence gives the number of toothpicks after n stages. A152968 (the first differences) gives the number of toothpicks added to the structure at n-th stage. %C A152998 For more information see A139250. (End) %H A152998 Seiichi Manyama, <a href="/A152998/b152998.txt">Table of n, a(n) for n = 0..8191</a> %H A152998 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 A152998 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 A152998 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a> %F A152998 a(n) = (A139250(n+1)-1)/2. %F A152998 From _Omar E. Pol_, Oct 01 2011: (Start) %F A152998 a(n) = A139250(n+1) - A153003(n) + A153000(n-1) - 1, if n >= 1. %F A152998 a(n) = A153003(n) - A153000(n-1), if n >= 1. %F A152998 a(n) = 2*A153000(n-1) + 1, if n >= 1. %F A152998 (End) %F A152998 a(n) = (A187220(n+2) - 3)/4. - _Omar E. Pol_, Feb 18 2013 %Y A152998 Cf. A139250, A139251, A152968. %Y A152998 Cf. A153000, A152978. %K A152998 nonn %O A152998 0,3 %A A152998 _Omar E. Pol_, Dec 19 2008, Dec 23 2008, Jan 02 2008