

A255283


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


2



1, 7, 7, 31, 7, 49, 31, 145, 7, 49, 49, 217, 31, 217, 145, 601, 7, 49, 49, 217, 49, 343, 217, 1015, 31, 217, 217, 961, 145, 1015, 601, 2551, 7, 49, 49, 217, 49, 343, 217, 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
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OFFSET

0,2


COMMENTS

This is the number of ON cells in a certain twodimensional 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.
This is the oddrule cellular automaton defined by OddRule 537 (see EkhadSloaneZeilberger "OddRule Cellular Automata on the Square Grid" link).


LINKS

Table of n, a(n) for n=0..63.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A MetaAlgorithm for Creating Fast Algorithms for Counting ON Cells in OddRule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, OddRule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
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: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata


FORMULA

This is the Run Length Transform of A255284.


EXAMPLE

Here is the neighborhood f:
[X, 0, X]
[X, X, 0]
[X, X, X]
which contains a(1) = 7 ON cells.
From Omar E. Pol, Feb 22 2015: (Start)
Written as an irregular triangle in which row lengths are the terms of A011782:
1;
7;
7, 31;
7, 49, 31, 145;
7, 49, 49, 217, 31, 217, 145, 601;
7, 49, 49, 217, 49, 343, 217, 1015, 31, 217, 217, 961, 145, 1015, 601, 2551;
...
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.
(End)
From Omar E. Pol, Mar 19 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
1;
..
7;
..
7;
31;
..........
7, 49;
31;
145;
......................
7, 49, 49, 217;
31, 217;
145;
601;
............................................
7, 49, 49, 217, 49, 343, 217, 1015;
31, 217, 217, 961;
145, 1015;
601;
2551;
.......................................................................................
7, 49, 49, 217, 49, 343, 217, 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;
...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
(End)


MATHEMATICA

(* f = A255284 *) f[n_] := If[EvenQ[n], 2^(2n+3)5*7^(n/2), 2^(2n+3)11*7^((n1)/2)]/3; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* JeanFrançois Alcover, Jul 12 2017 *)


CROSSREFS

Cf. A255284.
Sequence in context: A186142 A188274 A255281 * A140252 A095343 A286830
Adjacent sequences: A255280 A255281 A255282 * A255284 A255285 A255286


KEYWORD

nonn


AUTHOR

N. J. A. Sloane and Doron Zeilberger, Feb 19 2015


STATUS

approved



