A253068 - OEIS

%I #27 Nov 23 2017 02:41:48

%S 1,6,28,112,456,1816,7288,29112,116536,465976,1864248,7456312,

%T 29826616,119303736,477220408,1908870712,7635504696,30541975096,

%U 122167987768,488671776312,1954687454776,7818749120056,31274997878328,125099988717112,500399960460856,2001599830658616,8006399345004088,32025597335277112,128102389430586936

%N The subsequence A253066(2^n-1).

%C A253066 is the Run Length Transform of this sequence.

%H Colin Barker, <a href="/A253068/b253068.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_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8).

%F G.f.: (1+3x+4x^2)/((1-x)(1+2x)(1-4x)).

%F a(n) = ((-2)^n+4^(2+n)-8)/9. - _Colin Barker_, Jul 16 2015

%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 f25:=1/x+1+x+1/y+y/x+x*y;

%p OddCA2(f25,8);

%t LinearRecurrence[{3, 6, -8}, {1, 6, 28}, 29] (* _Jean-François Alcover_, Nov 23 2017 *)

%o (PARI) a(n) = ((-2)^n+4^(2+n)-8)/9 \\ _Colin Barker_, Jul 16 2015

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

%Y Cf. A253066.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 23 2015