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%I #28 Feb 24 2021 02:48:18
%S 0,1,3,5,9,9,9,13,25,21,9,13,25,25,25,37,73,57,9,13,25,25,25,37,73,61,
%T 25,37,73,73,73,109,217,165,9,13,25,25,25,37,73,61,25,37,73,73,73,109,
%U 217,169,25,37,73,73,73,109,217,181,73,109,217,217,217,325,649,489,9,13,25
%N Number of T-toothpicks added at n-th stage to the T-toothpick structure of A160172.
%C Essentially the first differences of A160172.
%C For further information see the Applegate-Pol-Sloane paper, chapter 11: T-shaped toothpicks. See also the figure 16 in the mentioned paper. - _Omar E. Pol_, Nov 18 2011
%C The numbers n in increasing order such that the triple [n, n, n] can be found here, give A199111. [Observed by _Omar E. Pol_, Nov 18 2011. Confirmed by _Alois P. Heinz_, Nov 21 2011]
%D David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191
%H David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H 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.], which is also available at <a href="http://arxiv.org/abs/1004.3036">arXiv:1004.3036v2</a>
%H 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 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F a(n) = (2/3)*(3^wt(n-1) + 3^wt(n-2))+1 (where wt is A000120), for n >= 3. - _N. J. A. Sloane_, Jan 01 2010
%e From _Omar E. Pol_, Feb 09 2010: (Start)
%e If written as a triangle:
%e 0;
%e 1;
%e 3;
%e 5;
%e 9,9;
%e 9,13,25,21;
%e 9,13,25,25,25,37,73,57;
%e 9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,165;
%e 9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109,217,181,73,109,217,217,217,325,649,489;
%e 9,13,25,25,25,37,73,61,25,37,73,73,73,109,217,169,25,37,73,73,73,109...
%e (End)
%t wt[n_] := DigitCount[n, 2, 1];
%t a[0] = 0; a[1] = 1; a[2] = 3; a[n_] := 2/3 (3^wt[n-1] + 3^wt[n-2]) + 1;
%t Table[a[n], {n, 0, 68}] (* _Jean-François Alcover_, Aug 18 2018, after _N. J. A. Sloane_ *)
%Y Cf. A139250, A139251, A160172, A160121, A160171, A161207, A161329, A172311.
%K nonn
%O 0,3
%A _Omar E. Pol_, Jun 01 2009
%E More terms from _N. J. A. Sloane_, Jan 01 2010
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