OFFSET
0,2
COMMENTS
Number of ON cells at generation n of 1-D CA defined by Rule 62. - N. J. A. Sloane, Aug 09 2014
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Illustration of first 20 generations of Rule 62
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,0,-1).
FORMULA
Comments from N. J. A. Sloane, Aug 11 2014: (Start)
As one can see from the illustration, there is a pattern that repeats every three steps on the left and every 12 steps on the right.
More precisely, let L(n) denote the number of N cells in the part of the diagram to the left of the central line. Then L(3t+1)=2t+1, L(3t+2)=L(3t+3)=2t+2, which is 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, ... (essentially A004523). This has g.f. (x+x^2)/((1-x)(1-x^3)).
Let R(n) denote the number of ON cells on the central axis and to the right of this axis. Then R(0) through R(5) = [1, 2, 1, 4, 2, 4] and thereafter R(12k+6) through R(12k+17) = 7k + [5, 5, 4, 8, 6, 7, 8, 9, 8, 11, 9, 11] for k = 0,1,2,... The g.f. for R(n) is (1+2*x+x^2+3*x^3-x^4+x^5)/((1-x^3)*(1-x^4)).
Combining these, we find that a(n) = L(n) + R(n) has the generating function that is given in the next line. (End)
G.f.: (2*x^5 + x^4 + 5*x^3 + 3*x^2 + 3*x + 1)/((1-x^3)*(1-x^4)). - Hans Havermann, May 26 2002
a(n+7) = a(n+4)+a(n+3)-a(n) with initial terms 1, 3, 3, 6, 5, 8, 9. - N. J. A. Sloane, Jan 31 2015
MATHEMATICA
CoefficientList[Series[(2 x^5 + x^4 + 5 x^3 + 3 x^2 + 3 x + 1)/((1 - x^3) (1 - x^4)), {x, 0, 80}], x] (* Vincenzo Librandi, Aug 10 2014 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Hans Havermann, May 26 2002
STATUS
approved