A sequence unique including differences

[Max Alekseyev]

There is also a number of related sequence that come to mind:

The sequence of the absolute first differences (is it A140779
mentioned in %Y?), their union with A140778, and the complement to
this union (i.e., numbers that appear neigher in A140778, nor in its
absolute first difference).

Regards,
Max

On Thu, May 29, 2008 at 5:12 PM,  <franktaw at netscape.net> wrote:
> I'm surprised that the following was not already in the OEIS:
>
> %I A140778
> %S A140778
> 1,3,7,12,18,8,17,28,13,27,43,19,39,60,22,45,70,26,55,85,31,63,96,34,69,10
> 5,37,77,
> 118,42,88,135,48,97,147,52,103,156,56,113,171,59,120,184,65,131,198,71,14
> 3,216,74,
> 149,227,79,159,240,82,165,249,86,175,265,91,183,276,94,192,291,101,203,30
> 7,106,
> 213,321,109,219,330,115,229,345,117,238,360,123,247,372,126,254,383,130,2
> 62,395,
> 134,270,407,138,277,417,144,285
> %N A140778 a(n) is the smallest positive integer such that no number occurs
> twice in the sequence and its absolute first differences.
> %C A140778 This sequence and its first differences include every positive
> integer (exactly once).
> %e A140778 For a(5), the sequence to that point is [1,3,7,12], with absolute
> differences [2,4,5].  The next number cannot be 6, because then 6 would be
> in
> both the sequence and the first differences.  Since all values smaller than
> 6
> are taken, the difference must be positive and at least 6.  A difference of
> 6
> works, a(5) = 18.
> %o A140778 (PARI) IsInList(v, k) =
> for(i=1,#v,if(v[i]==k,return(1)));return(0)
> IsInDiff(v, k) = for(i=2,#v,if(abs(v[i]-v[i-1])==k,return(1)));return(0)
> NextA140778(v)={
> local(i,d);
> if(#v==0,return(1));
> i=2;
> while(1,
>  d=abs(i-v[#v]);
>  if(!(i==d || IsInList(v,i) || IsInDiff(v,i) || IsInList(v,d) ||
> IsInDiff(v,d)), return(i));
>  i++)
> }
> v=[];for(i=1,100,v=concat(v,NextA140778(v)));v
> %Y A140778 Cf. A140779, A081145.
> %O A140778 1
> %K A140778 ,easy,nonn,
> %A A140778 Franklin T. Adams-Watters (FrankTAW at Netscape.net), May 29 2008
>
> Question: can an upper bound for a(n)/n be established?  If so, what is the
> least upper bound?  Up to n=100, the maximum is n=74, a(n) = 321, a(n)/n ~
> 4.338.
>
> The same question can be asked for A081145; the maximum for the values in
> the database is n=51, a(n) = 115, a(n)/n ~ 2.255.
>
> Franklin T. Adams-Watters
>



the OEIS that is relevant, in case you missed it: