|
%I #31 Nov 27 2017 02:49:17
%S 1,7,21,95,333,1319,4837,18447,68733,259447,972565,3661535,13756333,
%T 51754567,194586181,731919279,2752461533,10352254743,38932913525,
%U 146424889471,550683608589,2071066796007,7789015542949,29293584500047,110169505843517,414334209685687
%N The subsequence A253071(2^n-1).
%C A253071 is the Run Length Transform of this sequence.
%C A253072(2^k-1) = A050476(2^k-1), 0<=k<=3. This is just a coincidence, since it fails at m=4. - _Omar E. Pol_, Feb 01 2015; _N. J. A. Sloane_, Feb 20 2015
%H Colin Barker, <a href="/A253072/b253072.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, 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, 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, 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 (6,-5,-24,44,-8).
%F G.f.: -(-1-t+16*t^2-28*t^3+8*t^4)/(1-6*t+5*t^2+24*t^3-44*t^4+8*t^5).
%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*y)+1/x+1/y+y+x/y+x+x*y;
%p OddCA2(f25,8);
%t LinearRecurrence[{6, -5, -24, 44, -8}, {1, 7, 21, 95, 333}, 26] (* _Jean-François Alcover_, Nov 27 2017 *)
%o (PARI) Vec(-(8*x^4-28*x^3+16*x^2-x-1)/(8*x^5-44*x^4+24*x^3+5*x^2-6*x+1) + O(x^30)) \\ _Colin Barker_, Jul 16 2015
%Y Cf. A253067, A253068, A253071, A050476.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 31 2015
|
