login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A255283 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-x-y. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

This is the odd-rule cellular automaton defined by OddRule 537 (see Ekhad-Sloane-Zeilberger "Odd-Rule 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 Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.

Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, Odd-Rule 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^((n-1)/2)]/3; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 23 18:13 EDT 2019. Contains 321433 sequences. (Running on oeis4.)