A183004 -id:A183004 - cp's OEIS frontend

A183005 Number of toothpicks added at n-th stage to the toothpick structure of A183004.

0, 1, 4, 6, 8, 8, 16, 22, 16, 8, 16, 24, 24, 32, 56, 62, 32, 8, 16, 24, 24, 32, 56, 64, 40, 32, 56, 72, 80, 120, 176, 158, 64, 8, 16, 24, 24, 32, 56, 64, 40, 32, 56, 72, 80, 120, 176

Offset: 0

Omar E. Pol, Mar 27 2011

nonn

Essentially the first differences of A183004.

			If written as a triangle begins:
  0,
  1,
  4,
  6,8,
  8,16,22,16,
  8,16,24,24,32,56,62,32,
  8,16,24,24,32,56,64,40,32,56,72,80,120,176,158,64,

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. A139250, A139251, A182841, A183004.

A183126 Toothpick sequence with toothpicks connected by their endpoints.

Original entry on oeis.org

0, 1, 7, 23, 39, 79, 95, 135, 175, 287, 303, 343, 383, 495, 535, 647, 759, 1087, 1103, 1143, 1183, 1295, 1335, 1447, 1559, 1887, 1927, 2039, 2151, 2479, 2591, 2919, 3247, 4223, 4239, 4279, 4319, 4431, 4471, 4583, 4695

Offset: 0

Omar E. Pol, Mar 28 2011

nonn

On the infinite square grid we start with no toothpicks.
At stage 1 we place a single toothpick of length 1.
Rule: each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation.
The sequence gives the number of toothpicks after n stages. A183127 gives the number of toothpicks added at the n-th stage.

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, p. 31.
John W. Layman, Graphs of the toothpick configuration for generations 1-15
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A160172, A160410, A183004, A183127, A183148.

a[n_] := 7 + 4 (n - 2 + Sum[3^DigitCount[k, 2, 1], {k, n - 2}]); a[0] = 0; a[1] = 1; Array[a, 41, 0] (* Michael De Vlieger, Nov 02 2022 *)

From Nathaniel Johnston, Apr 06 2011: (Start)
a(n) = 7 + 4*(n-2 + Sum_{k=1..n-2}3^A000120(k)), n >= 2.
a(n) = 7 + 4*(n-2 + 3*A151920(n-3)), n >= 3.
a(1 + 2^n) = 2^(n+2)+4^(n+1)-1, n >= 0.
(End)

Terms a(0)-a(10) confirmed and terms a(11)-a(35) added by John W. Layman, Mar 30 2011

a(36)-a(40) from Nathaniel Johnston, Apr 06 2011

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.

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A183005 Number of toothpicks added at n-th stage to the toothpick structure of A183004.

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A183126 Toothpick sequence with toothpicks connected by their endpoints.

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

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