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A182838 H-toothpick sequence in the first quadrant starting with a D-toothpick placed on the diagonal [(0,1),(1,2)]. 5
0, 1, 3, 7, 11, 15, 21, 31, 39, 43, 49, 61, 77, 91, 105, 127, 143, 147, 153 (list; graph; refs; listen; history; text; internal format)



A H-toothpick sequence is a toothpick sequence on a square grid so that the toothpick structure resembles a honeycomb unfinished.

Using a square grid is simulated to be on a hexagonal net.

The structure has two types of elements: the classic toothpick with length 1 and the "D-toothpick"  with length 2^(1/2)=sqrt(2).

Toothpicks are placed in the vertical direction and D-toothpicks are placed in diagonal direction.

Each hexagon has area = 4.

The network looks like an elongated hexagonal lattice placed on the square grid so that all nodes of the hexagonal net coincide with some of the grid points of the square grid. Each node in the hexagonal network is represented with coordinates x,y.

The sequence gives the number of toothpicks and D-toothpicks after n steps. A182839 (first difference) gives the number added at the n-th stage.


Table of n, a(n) for n=0..18.

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

Index entries for sequences related to toothpick sequences


We start at stage 0 with no toothpicks.

At stage 1 we place a D-toothpick [(0,1),(1,2)], so a(1)=1.

At stage 2 we place a toothpick [(1,2),(1,3)] and a D-toothpick [(1,2),(2,1)], so a(2)=1+2=3.

At stage 3 we place 4 elements: a D-toothpick [(1,3),(0,4)], a D-toothpick [(1,3),(2,4)], a D-toothpick [(2,1),(3,2)] and a toothpick [(2,1),(2,0)], so a(3)=3+4=7. Etc.

The first hexagon appears in the structure after 4 stages.


Cf. A139250, A161206, A182632, A182634, A182839, A182840.

Sequence in context: A340699 A160802 A170888 * A327332 A163094 A022800

Adjacent sequences:  A182835 A182836 A182837 * A182839 A182840 A182841




Omar E. Pol, Dec 12 2010



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Last modified February 25 12:00 EST 2021. Contains 341606 sequences. (Running on oeis4.)