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%I #40 Feb 24 2021 02:48:19
%S 0,1,4,7,12,22,20,22,40,54,40,22,40,54,56,70,120,134,72,22,40,54,56,
%T 70,120,134,88,70,120,150,168,246,360,326,136,22,40,54,56,70,120,134,
%U 88,70,120,150,168,246,360,326,152,70,120,150,168,246,360,342,232,246,376,454,568,838,1032
%N First differences of A187210.
%C Number of Q-toothpicks added at n-th stage to the Q-toothpick structure of A187210.
%C For the connection with A139251, the first differences of the toothpick sequence A139250, see the Formula section. - _Omar E. Pol_, Apr 02 2016
%H Nathaniel Johnston, <a href="/A187211/b187211.txt">Table of n, a(n) for n = 0..177</a>
%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.]
%H Nathaniel Johnston, <a href="/A187211/a187211.c.txt">C script</a>
%H Nathaniel Johnston, <a href="http://www.nathanieljohnston.com/2011/03/the-q-toothpick-cellular-automaton/">The Q-Toothpick Cellular Automaton</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>
%F a(2^n + 2) = 16 + 8(2^(n-1) - 1), n >= 3. [_Nathaniel Johnston_, Mar 26 2011]
%F From _Omar E. Pol_, Apr 02 2016: (Start)
%F a(n) = floor(sqrt(2*n^3)), if 0<=n<=2 or n=6.
%F a(n) = 2*A139251(n-2) + A267699(n-2) + A267695(n-1), if 3<=n<=5 or n>=7.
%F (End)
%e Written as an irregular triangle the sequence begins:
%e 0;
%e 1;
%e 4;
%e 7;
%e 12;
%e 22, 20;
%e 22, 40, 54, 40;
%e 22, 40, 54, 56, 70, 120, 134, 72;
%e 22, 40, 54, 56, 70, 120, 134, 88, 70, 120, 150, 168, 246, 360, 326, 136;
%e ...
%e The rows of this triangle tend to A188156.
%e From _Omar E. Pol_, Apr 02 2016: (Start)
%e For n = 5 we have that A139251(5-2) = 4, A267699(5-2) = 7 and A267695(5-1) = 7, so a(5) = 2*4 + 7 + 7 = 22.
%e For n = 10 we have that A139251(10-2) = 8, A267699(10-2) = 20 and A267695(10-1) = 4, so a(10) = 2*8 + 20 + 4 = 40.
%e (End)
%e Starting from a(3) = 7 the row lengths of triangle are the terms of A011782. - _Omar E. Pol_, Apr 04 2016
%Y Cf. A011782, A139250, A139251, A172472, A182841, A187210, A187221, A267695, A267699.
%K nonn,tabf
%O 0,3
%A _Omar E. Pol_, Mar 07 2011
%E Terms after a(7) from _Nathaniel Johnston_, Mar 26 2011
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