A255283 - OEIS

%I #30 Jul 12 2017 05:22:33

%S 1,7,7,31,7,49,31,145,7,49,49,217,31,217,145,601,7,49,49,217,49,343,

%T 217,1015,31,217,217,961,145,1015,601,2551,7,49,49,217,49,343,217,

%U 1015,49,343,343,1519,217,1519,1015,4207,31,217,217,961,217,1519,961,4495,145,1015,1015,4495,601,4207,2551,10351

%N Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-x-y.

%C This is the number of ON cells in a certain two-dimensional cellular automaton in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there were an odd number of ON cells in the neighborhood at the previous generation.

%C This is the odd-rule cellular automaton defined by OddRule 537 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

%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>

%F This is the Run Length Transform of A255284.

%e Here is the neighborhood f:

%e [X, 0, X]

%e [X, X, 0]

%e [X, X, X]

%e which contains a(1) = 7 ON cells.

%e From _Omar E. Pol_, Feb 22 2015: (Start)

%e Written as an irregular triangle in which row lengths are the terms of A011782:

%e 1;

%e 7;

%e 7, 31;

%e 7, 49, 31, 145;

%e 7, 49, 49, 217, 31, 217, 145, 601;

%e 7, 49, 49, 217, 49, 343, 217, 1015, 31, 217, 217, 961, 145, 1015, 601, 2551;

%e ...

%e Right border gives: 1, 7, 31, 145, 601, 2551, ... This is simply a restatement of the theorem that this sequence is the Run Length Transform of A255284.

%e (End)

%e From _Omar E. Pol_, Mar 19 2015: (Start)

%e Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:

%e 1;

%e ..

%e 7;

%e ..

%e 7;

%e 31;

%e ..........

%e 7,     49;

%e 31;

%e 145;

%e ......................

%e 7,     49,   49,  217;

%e 31,   217;

%e 145;

%e 601;

%e ............................................

%e 7,     49,   49,  217,  49,  343, 217, 1015;

%e 31,   217,  217,  961;

%e 145, 1015;

%e 601;

%e 2551;

%e .......................................................................................

%e 7,     49,   49,  217,  49,  343, 217, 1015, 49, 343, 343, 1519, 217, 1519, 1015, 4207;

%e 31,   217,  217,  961, 217, 1519, 961, 4495;

%e 145, 1015, 1015, 4495;

%e 601, 4207;

%e 2551;

%e 10351;

%e ...

%e Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).

%e (End)

%t (* f = A255284 *) f[n_] := If[EvenQ[n], 2^(2n+3)-5*7^(n/2), 2^(2n+3)-11*7^((n-1)/2)]/3; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* _Jean-François Alcover_, Jul 12 2017 *)

%Y Cf. A255284.

%K nonn

%O 0,2

%A _N. J. A. Sloane_ and _Doron Zeilberger_, Feb 19 2015