A246030 a(n) = (5*2^(2*n)+(-2)^(n+1))/3.

1, 8, 24, 112, 416, 1728, 6784, 27392, 109056, 437248, 1746944, 6991872, 27959296, 111853568, 447381504, 1789591552, 7158235136, 28633202688, 114532286464, 458130194432, 1832518680576, 7330078916608, 29320307277824, 117281245888512, 469124949999616, 1876499867107328, 7505999334211584

Offset: 0

Author

N. J. A. Sloane, Aug 15 2014

Keywords

• nonn
• easy

Comments

Essentially 8 times A083424.

Equals A160239(2^n). - N. J. A. Sloane, Oct 04 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
• Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
• N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
• N. J. A. Sloane, Illustration for a(4) = 416.
• N. J. A. Sloane, Illustration for a(5) = 1728.
• N. J. A. Sloane, Illustration for a(6) = 6784.
• N. J. A. Sloane, Illustration for a(7) = 27392 (tiff).
• N. J. A. Sloane, Illustration for a(7) = 27392 (png).
• Index entries for sequences related to cellular automata
• Index entries for linear recurrences with constant coefficients, signature (2,8).

Crossrefs

Cf. A001045, A083424, A160239.

Programs

Magma

I:=[1,8]; [n le 2 select I[n] else 2*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 20 2015

Mathematica

CoefficientList[Series[(1 + 6 x) / (1 - 2 x - 8 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 20 2015 *)

Formula

a(0)=1, a(1)=8; thereafter a(n)=2*a(n-1)+8*a(n-2).

G.f.: (1+6*x)/(1-2*x-8*x^2).

a(n) = A001045(n+2)^2 - A001045(n)^2. - J. Conrad, Apr 05 2023