A183126 -id:A183126 - cp's OEIS frontend

A183127 Number of toothpicks added at n-th stage to the toothpick structure of A183126.

0, 1, 6, 16, 16, 40, 16, 40, 40, 112, 16, 40, 40, 112, 40, 112, 112, 328, 16, 40, 40, 112, 40, 112, 112, 328, 40, 112, 112, 328, 112, 328, 328, 976, 16, 40, 40, 112, 40, 112, 112, 328, 40, 112, 112, 328, 112, 328, 328, 976, 40, 112, 112, 328, 112, 328, 328, 976

Offset: 0

Omar E. Pol, Mar 28 2011

nonn

Essentially the first differences of A183126.

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.

Cf. A000120, A139250, A139251, A160173, A161411, A183126, A183149.

a[n_] := 4*(1 + 3^DigitCount[n-2, 2, 1]); a[0] = 0; a[1] = 1; a[2] = 6; Array[a, 100, 0] (* Amiram Eldar, Aug 01 2023 *)

a(n) = 4*(1 + 3^A000120(n-2)), n >= 3. - Nathaniel Johnston, Apr 06 2011

a(0)-a(10) confirmed and a(11)-a(35) added by John W. Layman, Mar 30 2011
a(36)-a(47) from Nathaniel Johnston, Apr 06 2011
More terms from Amiram Eldar, Aug 01 2023

A183148 Toothpick sequence on the semi-infinite square grid with toothpicks connected by their endpoints.

Original entry on oeis.org

0, 1, 4, 13, 22, 43, 52, 73, 94, 151, 160, 181, 202, 259, 280, 337, 394, 559, 568, 589, 610, 667, 688, 745, 802, 967, 988, 1045, 1102, 1267, 1324, 1489, 1654, 2143, 2152, 2173, 2194, 2251, 2272, 2329, 2386, 2551, 2572, 2629

Offset: 0

Omar E. Pol, Mar 28 2011, Apr 03 2011

nonn

On the semi-infinite square grid we start with no toothpicks.
At stage 1 we place a single toothpick of length 1 which has one of its endpoints on the straight line.
New generations of toothpicks are added according to these rules: each exposed endpoint of toothpicks of the old generation must be touched by the 3 endpoints of three toothpicks of the new generation. Effectively these three toothpicks look like a T-toothpick (see A160172). The straight line that delimits the square grid acts like an impenetrable "absorbing" boundary: toothpicks may touch this line with at most one of their endpoints; these endpoints are not "exposed."
The sequence gives the number of toothpicks in the toothpick structure after n-th stage. The first differences (A183149) give the number of toothpicks added at n-th stage.

			At stage 1 place an orthogonal toothpick with one of its endpoints on the infinite straight line, so a(1) = 1. There is only one exposed endpoint.
At stage 2 place 3 toothpicks such that the structure looks like a cross, so a(2) = 1+3 = 4. There are 3 exposed endpoints.
At stage 3 place 9 toothpicks, so a(3) = 4+9 = 13. There are 3 exposed endpoints.
At stage 4 place 9 toothpicks, so a(4) = 13+9 = 22. There are 7 exposed endpoints.

Michael De Vlieger, Table of n, a(n) for n = 0..1000
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.]
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 31-32.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A147562, A151920, A183004, A183060, A183126, A183149.

s[n_] := 1 + 4 Sum[3^(DigitCount[k, 2, 1] - 1), {k, n - 1}]; {0}~Join~Array[3 (# + (s[#] - 1)/2) + 1 &, 43, 0] (* Michael De Vlieger, Nov 02 2022 *)

a(n) = 3*A183060(n-1) + 1.

cp's OEIS Frontend

A183127 Number of toothpicks added at n-th stage to the toothpick structure of A183126.

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