cp's OEIS Frontend

A253064 Number of odd terms in f^n, where f = 1/x + 1 + x + y.

%I A253064 #35 Jul 18 2023 03:35:21
%S A253064 1,4,4,12,4,16,12,40,4,16,16,48,12,48,40,128,4,16,16,48,16,64,48,160,
%T A253064 12,48,48,144,40,160,128,416,4,16,16,48,16,64,48,160,16,64,64,192,48,
%U A253064 192,160,512,12,48,48,144,48,192,144,480,40,160,160,480,128,512,416
%N A253064 Number of odd terms in f^n, where f = 1/x + 1 + x + y.
%C A253064 This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.
%C A253064 This is the odd-rule cellular automaton defined by OddRule 017 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - _N. J. A. Sloane_, Feb 25 2015
%H A253064 Alois P. Heinz, <a href="/A253064/b253064.txt">Table of n, a(n) for n = 0..8191</a>
%H A253064 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 A253064 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 A253064 Po-Yi Huang and Wen-Fong Ke, <a href="https://arxiv.org/abs/2307.07733">Sequences Derived from The Symmetric Powers of {1,2,...,k}</a>, arXiv:2307.07733 [math.CO], 2023.
%H A253064 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 A253064 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 A253064 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A253064 This is the Run Length Transform of A087206.
%e A253064 Here is the neighborhood f:
%e A253064 [0, X, 0]
%e A253064 [X, X, X]
%e A253064 which contains a(1) = 4 ON cells.
%p A253064 C:=f->subs({x=1, y=1}, f);
%p A253064 # Find number of ON cells in CA for generations 0 thru M defined by rule
%p A253064 # that cell is ON iff number of ON cells in nbd at time n-1 was odd
%p A253064 # where nbd is defined by a polynomial or Laurent series f(x, y).
%p A253064 OddCA:=proc(f, M) global C; local n, a, i, f2, p;
%p A253064 f2:=simplify(expand(f)) mod 2;
%p A253064 a:=[]; p:=1;
%p A253064 for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
%p A253064 lprint([seq(a[i], i=1..nops(a))]);
%p A253064 end;
%p A253064 f:=1/x+1+x+y;
%p A253064 OddCA(f, 130);
%t A253064 f[n_] := 2^n*Fibonacci[n+2]; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 62}] (* _Jean-François Alcover_, Jul 11 2017 *)
%Y A253064 Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035.
%Y A253064 Cf. A087206.
%K A253064 nonn
%O A253064 0,2
%A A253064 _N. J. A. Sloane_, Jan 26 2015