%I #16 Feb 24 2021 02:48:19

%S 0,1,5,21,37,53,117,197,261,405,565,789,965,1221,1541,1941,2453,2933,

%T 3621,4389,5093,5909,6805,7925,9093,10629,12197,14133,15733,17717,

%U 19493,21605,23909,26453,29109,32117,35013,38085,41285

%N Number of toothpicks after n stages of 3-D toothpick structure defined in Comments.

%C We are in 3-D, and we are placing ordinary toothpicks, as in A139250.

%C We start with one toothpick in the z direction

%C We place toothpicks at any free end, as in A139250.

%C We always place new toothpicks in pairs, two perpendicular toothpicks that are perpendicular to the original toothpick

%C The toothpicks are always in 2 out of the 3 (x, y or z) directions.

%C The initial values are as follows (this should be checked!):

%C n:.0..1..2..3..4..5..6

%C ----------------------------

%C x..0..0..2..4..8..4.24 (Number added in x direction)

%C y..0..0..2..4..8..4.24 (Number added in y direction)

%C z..0..1..0..8..0..8.16 (Number added in z direction)

%C ----------------------------

%C ...0..1..4.16.16.16.64 (Total number added at n-th stage, A170876)

%C ----------------------------

%C a..0..1..5.21.37.53.117 (Total so far, this sequence)

%C ----------------------------

%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 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 R. J. Mathar, <a href="/A170876/a170876.cc">C++ program</a>

%H R. J. Mathar, <a href="/A170876/a170876_2.eps">View after stage 1</a>

%H R. J. Mathar, <a href="/A170876/a170876_3.eps">View after stage 2</a>

%H R. J. Mathar, <a href="/A170876/a170876_4.eps">View after stage 3</a>

%H R. J. Mathar, <a href="/A170876/a170876_5.eps">View after stage 4</a>

%H R. J. Mathar, <a href="/A170876/a170876_6.eps">View after stage 5</a>

%H R. J. Mathar, <a href="/A170876/a170876_7.eps">View after stage 6</a>

%H R. J. Mathar, <a href="/A170876/a170876_8.eps">View after stage 7</a>

%H R. J. Mathar, <a href="/A170876/a170876_9.eps">View after stage 8</a>

%H R. J. Mathar, <a href="/A170876/a170876_10.eps">View after stage 9</a>

%H R. J. Mathar, <a href="/A170876/a170876_11.eps">View after stage 10</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp053.jpg">Illustration of initial terms</a>

%e At stage 2 we have a horizontal cross, a vertical toothpick then another horizontal cross, for a total of 5 toothpicks.

%e 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 Cf. A139250, A170875 (first differences), A160160, A160170. For another version see A170837.

%K nonn

%O 0,3

%A _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