A253067 - OEIS

%I #32 Nov 23 2017 02:41:35

%S 1,5,17,65,229,813,2945,10513,37701,135261,484609,1737665,6229413,

%T 22330829,80057281,286996657,1028861637,3688409853,13222664897,

%U 47402353633,169934149285,609201913325,2183946525185,7829295473489,28067476697413,100619943566493

%N The subsequence A253065(2^n-1).

%C A253065 is the Run Length Transform of this sequence.

%H Colin Barker, <a href="/A253067/b253067.txt">Table of n, a(n) for n = 0..1000</a>

%H Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, <a href="http://arxiv.org/abs/1503.01796">A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata</a>, arXiv:1503.01796 [math.CO], 2015; see also the <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.html">Accompanying Maple Package</a>.

%H Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, <a href="http://arxiv.org/abs/1503.04249">Odd-Rule Cellular Automata on the Square Grid</a>, arXiv:1503.04249 [math.CO], 2015.

%H 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: <a href="https://vimeo.com/119073818">Part 1</a>, <a href="https://vimeo.com/119073819">Part 2</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 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,13,6,8).

%F G.f.: (1+2*x)*(1+2*x+3*x^2+4*x^3)/(1-x-5*x^2-13*x^3-6*x^4-8*x^5).

%F Examination of the roots of the denominator shows that the ratio of successive terms approaches 3.5849301...

%p OddCA2:=proc(f,M) local n,a,i,f2,g,p;

%p    f2:=simplify(expand(f)) mod 2;

%p    p:=1; g:=f2;

%p    for n from 1 to M do p:=expand(p*g) mod 2; print(n,nops(p)); g:=expand(g^2) mod 2; od:

%p    return;

%p end;

%p f24:=1/x+1+x+x/y+x*y;

%p OddCA2(f24,8);

%t LinearRecurrence[{1, 5, 13, 6, 8}, {1, 5, 17, 65, 229}, 26] (* _Jean-François Alcover_, Nov 23 2017 *)

%o (PARI) Vec(-(2*x+1)*(4*x^3+3*x^2+2*x+1)/(8*x^5+6*x^4+13*x^3+5*x^2+x-1) + O(x^30)) \\ _Colin Barker_, Jul 16 2015

%Y Cf. A253065.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jan 27 2015