%I #43 Jul 26 2024 21:16:31

%S 1,4,16,24,32,40,48,56,64,72,80,88,96,104,112,120,128,136,144,152,160,

%T 168,176,184,192,200,208,216,224,232,240,248,256,264,272,280,288,296,

%U 304,312,320,328,336,344,352,360,368,376,384,392,400,408,416,424,432

%N Expansion of (1+2*x+9*x^2-4*x^3)/(1-x)^2.

%C Also the coordination sequence for a point of degree 4 in the tiling of the Euclidean plane by right triangles (with angles Pi/2, Pi/4, Pi/4). These triangles are fundamental regions for the Coxeter group (2,4,4). In the notation of Conway et al. 2008 this is the tiling *442. The coordination sequence for a point of degree 8 is given by A022144. - _N. J. A. Sloane_, Dec 28 2015

%C First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood. Initialized with a single black (ON) cell at stage zero. - _Robert Price_, May 28 2016

%D J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5. See p. 191.

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

%H Colin Barker, <a href="/A234275/b234275.txt">Table of n, a(n) for n = 0..1000</a>

%H J. Choi, N. Pippenger, <a href="http://arxiv.org/abs/1310.1357">Counting the Angels and Devils in Escher's Circle Limit IV</a>, arXiv preprint arXiv:1310.1357, 2013.

%H Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.

%H N. J. A. Sloane, <a href="/A296368/a296368_2.png">Overview of coordination sequences of Laves tilings</a> [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(n) = A022144(n), n>1. - _R. J. Mathar_, Jan 11 2014

%F From _Colin Barker_, Jul 10 2015: (Start)

%F a(n) = 8*n, n>1.

%F a(n) = 2*a(n-1) - a(n-2) for n>3.

%F G.f.: -(4*x^3-9*x^2-2*x-1) / (x-1)^2.

%F (End)

%t Join[{1, 4}, LinearRecurrence[{2, -1}, {16, 24}, 60]] (* _Jean-François Alcover_, Jan 08 2019 *)

%o (PARI) Vec(-(4*x^3-9*x^2-2*x-1)/(x-1)^2 + O(x^100)) \\ _Colin Barker_, Jul 10 2015

%Y Cf. A022144, A008590.

%Y For partial sums see A265056.

%Y List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Dec 24 2013