A185587 Irregular triangle read by rows: row n gives a list of the lengths of the free spaces at the n-th stage in a Rule 18 cellular automaton.

1, 3, 1, 1, 1, 7, 1, 5, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 15, 1, 13, 1, 3, 11, 3, 1, 1, 1, 9, 1, 1, 1, 7, 7, 7, 1, 5, 1, 5, 1, 5, 1, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 31, 1, 29, 1, 3, 27, 3, 1, 1, 1, 25, 1, 1, 1, 7, 23, 7, 1, 5, 1, 21, 1, 5, 1, 3, 3, 3, 19, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 15, 15, 15, 1, 13, 1, 13, 1, 13, 1, 3, 11, 3, 11, 3, 11, 3

Offset

1,2

Comments

a(n) is the size of the n^th free space inside the development of a Rule 18 CA (related to logical XOR) on a tape started with a single 1.

Links

Benjamin Heiland, Table of n, a(n) for n = 1..154474

Example

Cellular automaton defined by Rule 18 (with sizes in blank space):
(x is a 1 in the automaton, the numbers are the sizes of white spaces)
                x          ->
               x1x         ->1
              x 3 x        ->3
             x1x1x1x       ->1,1,1
            x   7   x      ->7
           x1x  5  x1x     ->1,5,1
          x 3 x 3 x 3 x    ->3,3,3
         x1x1x1x1x1x1x1x   ->1,1,1,1,1,1,1
and so on.

Prog

(C)
#include<stdio.h>
#include<stdlib.h>
int main(){
int inumgen;                   //number of generations
int *iacurrentgen;             //current generation
int *ialastgen;                //last genereation (to calculate currentgen)
int i=0;                       //loop counter
int j=0;                       //another loop counter
int nullcount=0;               //used to determinate whitespace size
int a;                         //a for XOR to get new value
int b;                         //b for XOR to get new value
iacurrentgen=(int*)calloc(1, sizeof(int));
ialastgen=(int*)calloc(1, sizeof(int));
ialastgen[0]=1;
printf("Calculating A185587\n");
printf("please enter number of generations\n");
printf("note that the number of sequence elements per Generation is fluctuating.\n");
scanf("%d", &inumgen);
i++;                           //we start at generation1 , not at offset.
while(i<=inumgen){
  iacurrentgen=(int*)realloc(iacurrentgen, ((i*2+1)*sizeof(int)));
  while(j<(i*2+1)){
   if((j-2)<0)
    a=0;
   else
    a=ialastgen[j-2];
   if(j>((i-1)*2))
    b=0;
   else
    b=ialastgen[j];
   iacurrentgen[j]=(a||b)&&!(a&&b);      //(a||b)&&!(a&&b)=aXORb
   j++;
  }
  j=0;
  ialastgen=(int*)realloc(ialastgen, ((i*2+1)*sizeof(int)));
  while(j<=i*2){
   ialastgen[j]=iacurrentgen[j];
   if(iacurrentgen[j]==1){
    if(nullcount!=0){
     printf("%d, ", nullcount);
     nullcount=0;
    }
   }
   if(iacurrentgen[j]==0){
    nullcount++; }
   j++;
  }
  j=0;
  printf("\n");
  i++;
}
}

Crossrefs

Cf. A070886 (Rule 90 = Rule 18, starting with 1).

Sequence in context: A293682 A276651 A374126 * A068845 A324910 A257100

Adjacent sequences: A185584 A185585 A185586 * A185588 A185589 A185590

Keyword

nonn,tabf

Author

Benjamin Heiland, Feb 04 2011

Status

approved