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A170876 Number of toothpicks after n stages of 3-D toothpick structure defined in Comments. 5
0, 1, 5, 21, 37, 53, 117, 197, 261, 405, 565, 789, 965, 1221, 1541, 1941, 2453, 2933, 3621, 4389, 5093, 5909, 6805, 7925, 9093, 10629, 12197, 14133, 15733, 17717, 19493, 21605, 23909, 26453, 29109, 32117, 35013, 38085, 41285 (list; graph; refs; listen; history; text; internal format)



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

We start with one toothpick in the z direction

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

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

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

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



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

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

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


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


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



Table of n, a(n) for n=0..38.

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. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

R. J. Mathar, C++ program

R. J. Mathar, View after stage 1

R. J. Mathar, View after stage 2

R. J. Mathar, View after stage 3

R. J. Mathar, View after stage 4

R. J. Mathar, View after stage 5

R. J. Mathar, View after stage 6

R. J. Mathar, View after stage 7

R. J. Mathar, View after stage 8

R. J. Mathar, View after stage 9

R. J. Mathar, View after stage 10

Omar E. Pol, Illustration of initial terms


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

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.


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

Sequence in context: A063575 A302873 A170837 * A341198 A038844 A303521

Adjacent sequences:  A170873 A170874 A170875 * A170877 A170878 A170879




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



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Last modified July 23 17:41 EDT 2021. Contains 346259 sequences. (Running on oeis4.)