A183126 - OEIS

%I #34 Nov 02 2022 11:54:38

%S 0,1,7,23,39,79,95,135,175,287,303,343,383,495,535,647,759,1087,1103,

%T 1143,1183,1295,1335,1447,1559,1887,1927,2039,2151,2479,2591,2919,

%U 3247,4223,4239,4279,4319,4431,4471,4583,4695

%N Toothpick sequence with toothpicks connected by their endpoints.

%C On the infinite square grid we start with no toothpicks.

%C At stage 1 we place a single toothpick of length 1.

%C Rule: each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of three toothpicks of new generation.

%C The sequence gives the number of toothpicks after n stages. A183127 gives the number of toothpicks added at the n-th stage.

%H Michael De Vlieger, <a href="/A183126/b183126.txt">Table of n, a(n) for n = 0..1000</a>

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]

%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 31.

%H John W. Layman, <a href="https://personal.math.vt.edu/layman/sequences/A183126.htm">Graphs of the toothpick configuration for generations 1-15</a>

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%F From _Nathaniel Johnston_, Apr 06 2011: (Start)

%F a(n) = 7 + 4*(n-2 + Sum_{k=1..n-2}3^A000120(k)), n >= 2.

%F a(n) = 7 + 4*(n-2 + 3*A151920(n-3)), n >= 3.

%F a(1 + 2^n) = 2^(n+2)+4^(n+1)-1, n >= 0.

%F (End)

%t 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 *)

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

%K nonn

%O 0,3

%A _Omar E. Pol_, Mar 28 2011

%E Terms a(0)-a(10) confirmed and terms a(11)-a(35) added by _John W. Layman_, Mar 30 2011

%E a(36)-a(40) from _Nathaniel Johnston_, Apr 06 2011