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A162795 Total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.

%I A162795 #112 Aug 31 2021 02:42:32
%S A162795 1,5,9,21,25,37,53,85,89,101,117,149,165,201,261,341,345,357,373,405,
%T A162795 421,457,517,597,613,649,709,793,853,965,1173,1365,1369,1381,1397,
%U A162795 1429,1445,1481,1541,1621,1637,1673,1733,1817,1877,1989,2197,2389,2405,2441,2501
%N A162795 Total number of toothpicks in the toothpick structure A139250 that are parallel to the initial toothpick, after n odd rounds.
%C A162795 Partial sums of A162793.
%C A162795 Also, total number of ON cells at stage n of the two-dimensional cellular automaton defined as follows: replace every "vertical" toothpick of length 2 with a centered unit square "ON" cell, so we have a cellular automaton which is similar to both A147562 and A169707 (this is the "one-step bishop" version). For the "one-step rook" version we use toothpicks of length sqrt(2), then rotate the structure 45 degrees and then replace every toothpick with a unit square "ON" cell. For the illustration of the sequence as a cellular automaton we now have three versions: the original version with toothpicks, the one-step rook version and one-step bishop version. Note that the last two versions refer to the standard ON cells in the same way as the two versions of A147562 and the two versions of A169707. It appears that the graph of this sequence lies between the graphs of A147562 and A169707. Also, it appears that this sequence shares infinitely many terms with both A147562 and A169707, see Formula section and Example section. - _Omar E. Pol_, Feb 20 2015
%C A162795 It appears that this is also a bisection (the odd terms) of A255747.
%H A162795 Michel Marcus, <a href="/A162795/b162795.txt">Table of n, a(n) for n = 1..10000</a>
%H A162795 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 A162795 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 A162795 It appears that a(n) = A147562(n) = A169707(n), if n is a term of A048645, otherwise A147562(n) < a(n) < A169707(n). - _Omar E. Pol_, Feb 20 2015
%F A162795 It appears that a(n) = (A169707(2n) - 1)/4 = A255747(2n-1). - _Omar E. Pol_, Mar 07 2015
%F A162795 a(n) = 1 + 4*A255737(n-1). - _Omar E. Pol_, Mar 08 2015
%e A162795 From _Omar E. Pol_, Feb 18 2015: (Start)
%e A162795 Written as an irregular triangle T(j,k), k>=1, in which the row lengths are the terms of A011782:
%e A162795     1;
%e A162795     5;
%e A162795     9, 21;
%e A162795    25, 37, 53, 85;
%e A162795    89,101,117,149,165,201,261,341;
%e A162795   345,357,373,405,421,457,517,597,613,649,709,793,853,965,1173,1365;
%e A162795   ...
%e A162795 The right border gives the positive terms of A002450.
%e A162795 (End)
%e A162795 It appears that T(j,k) = A147562(j,k) = A169707(j,k), if k is a power of 2, for example: it appears that the three mentioned triangles only share the elements of the columns 1, 2, 4, 8, 16, ... - _Omar E. Pol_, Feb 20 2015
%Y A162795 Cf. A002450, A048645, A139250, A139251, A147562, A153000, A159791, A159792, A160164, A160552, A162793, A162794, A162796, A162797, A169707, A255263, A255264, A255747.
%K A162795 nonn
%O A162795 1,2
%A A162795 _Omar E. Pol_, Jul 14 2009
%E A162795 More terms from _N. J. A. Sloane_, Dec 28 2009