OFFSET
0,2
COMMENTS
The 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, initiated with a single black (ON) cell.- Robert Price, May 28 2016
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From Colin Barker, Jan 01 2016: (Start)
a(n) = 4*n^2+4*n-3 for n>0.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3.
G.f.: (1+2*x+9*x^2-4*x^3) / (1-x)^3.
(End)
MATHEMATICA
Accumulate[LinearRecurrence[{2, -1}, {1, 4, 16, 24}, 60]] (* or *) LinearRecurrence[{3, -3, 1}, {1, 5, 21, 45}, 60] (* Harvey P. Dale, Sep 22 2024 *)
PROG
(PARI) Vec((1+2*x+9*x^2-4*x^3)/(1-x)^3 + O(x^100)) \\ Colin Barker, Jan 01 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 28 2015
STATUS
approved