Curious relation: (Where A121561(n) = 1) == A093515

[Ferenc Adorjan]

Dear Maximilian,


> I noticed that the rule as applied here contradicts the definition at
> http://mathworld.wolfram.com/Rule110.html, since here  the sequence is
> shifted to the right by one bit after applying the ruler transform.
> Would this be done for any ruler ? (especially e.g. the mirror image
> of 110, 124 ?)

Yes, I apply a modified definition (corresponding to the Remark 8. in the
referred paper) to establish a correct mapping from (0,1) to (0,1). However,
this shifting does not modify the characteristic behaviour of the "rule"-s.

>
>
> A PARI program corresponding to the rule as used here is:
> binruletrans(v,r)={ local(t=0);
>  r=vecextract(binary(r+256),"-1..-8"); vector(#v,i,r[1+t=t%4*2+v[i]])}
>

Certainly, the PARI code I gave is not efficient. And yesterday I sas
already too tired to check the observation upto high indices.

>
> while the program reproducing the pattern at
> http://mathworld.wolfram.com/Rule110.html
> would be
> binruletrans2(v,r)={ local(t=v[1]); r=vecextract(binary(r+256),"-1..-8");
>  concat(vector(#v-1,i,r[1+t=t%4*2+v[i+1]]),r[1+t%4*2])}
>
> Anyway, Wilson's observation seems correct up to quite many digits.
>
> It can be checked with the more efficient (I believe) following
> implementation:
>
> \\ transform seq v by rule r without making detour via vector of
> bin.digits
> seqruletrans(v,r)={local(c=1,t); r=vecextract(binary(r+256),"-1..-8");
>  v=List(vector(v[#v],i,v[c]<i & c++; if(r[1+t=t%4*2+(v[c]==i)],i)));
>  listsort(v,1); t=Vec(v);
>  if( v[1],t=Vec[v], t=vecextract( Vec(v), "^1")); listkill(v); t}
>
> and (using MaxAl's observation:
> A121561_is_1(n) = vector(n+n=0,i,while(!isprime(n++)&!isprime(n-1),);n)
>
> one can check 10^4 digits in a fraction of a second:
>
> seqruletrans( vector(10^4,i,prime(i)),110);
> if( % - A121561_is_1(#%), wrong, true)
>
> I'm sure MaxAl will soon give the proof...
>

I am expecting (and looking forward to) it, also...
Regards to all
Ferenc

>
> Maximilian
>
>
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