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

0,3

Comments

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.

Links

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

Formula

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)

Mathematica

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 *)

Crossrefs

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

Sequence in context: A227064 A143030 A031043 * A213632 A031095 A319050

Adjacent sequences: A183123 A183124 A183125 * A183127 A183128 A183129

Keyword

nonn

Author

Omar E. Pol, Mar 28 2011

Extensions

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

Status

approved