OFFSET
0,2
LINKS
Colin Barker, 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 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (5,-4,-4).
FORMULA
G.f.: (1-x)*(1+2*x) / ((1-2*x)*(1-3*x-2*x^2)).
From Colin Barker, Feb 03 2017: (Start)
a(n) = (2^(-n)*(-17*4^n + (17-5*sqrt(17))*(3-sqrt(17))^n + (3+sqrt(17))^n*(17+5*sqrt(17)))) / 17.
a(n) = 5*a(n-1) - 4*a(n-2) - 4*a(n-3) for n>2.
(End)
MATHEMATICA
A255296[n_] := SeriesCoefficient[-(2*t + 1)*(t - 1)/((-1 + 2*t)*(2*t^2 + 3*t - 1)), {t, 0, n}]; Array[A255296, 24, 0] (* JungHwan Min, Sep 29 2016 *)
A255296L[n_] := CoefficientList[Series[-(2*t + 1)*(t - 1)/((-1 + 2*t)*(2*t^2 + 3*t - 1)), {t, 0, n}], t]; A255296L[23] (* JungHwan Min, Sep 29 2016 *)
LinearRecurrence[{5, -4, -4}, {1, 6, 24}, 30] (* Vincenzo Librandi, Feb 04 2017 *)
PROG
(PARI) Vec((1-x)*(1+2*x) / ((1-2*x)*(1-3*x-2*x^2)) + O(x^30)) \\ Colin Barker, Feb 03 2017
(Magma) I:=[1, 6, 24]; [n le 3 select I[n] else 5*Self(n-1)-4*Self(n-2)-4*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 04 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane and Doron Zeilberger, Feb 23 2015
STATUS
approved