A183127 -id:A183127 - cp's OEIS frontend

A183126 Toothpick sequence with toothpicks connected by their endpoints.

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

A183149 Number of toothpicks added at n-th stage to the toothpick structure of A183148.

Original entry on oeis.org

0, 1, 3, 9, 9, 21, 9, 21, 21, 57, 9, 21, 21, 57, 21, 57, 57, 165, 9, 21, 21, 57, 21, 57, 57, 165, 21, 57, 57, 165, 57, 165, 165, 489, 9, 21, 21, 57, 21, 57, 57, 165, 21, 57, 57, 165, 57, 165, 165, 489, 21, 57, 57, 165, 57, 165

Offset: 0

Omar E. Pol, Mar 28 2011, Apr 02 2011

nonn

Essentially the first differences of A183148.

			If written as a triangle begins:
0,
1,
3,
9,
9,21,
9,21,21,57,
9,21,21,57,21,57,57,165,
9,21,21,57,21,57,57,165,21,57,57,165,57,165,165,489,

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, A147582, A147610, A160173, A161411, A183061, A183127, A183148.

a(n) = 3*A183061(n-1), for n >=2

cp's OEIS Frontend

A183126 Toothpick sequence with toothpicks connected by their endpoints.

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