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A140778 a(n) is the smallest positive integer such that no number occurs twice in the union of the sequence and its absolute first differences.

%I A140778 #13 Dec 30 2024 01:42:13
%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,
%T A140778 69,105,37,77,118,42,88,135,48,97,147,52,103,156,56,113,171,59,120,
%U A140778 184,65,131,198,71,143,216,74,149,227,79,159,240,82,165,249,86,175,265,91,183
%N A140778 a(n) is the smallest positive integer such that no number occurs twice in the union of the sequence and its absolute first differences.
%C A140778 This sequence and its first differences include every positive integer (exactly once).
%H A140778 Alois P. Heinz, <a href="/A140778/b140778.txt">Table of n, a(n) for n = 1..10000</a>
%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.
%p A140778 b:= proc() false end:
%p A140778 a:= proc(n) option remember; local k;
%p A140778       if n=1 then b(1):= true; 1
%p A140778     else for k while b(k) or (t-> b(t) or t=k)(abs(a(n-1)-k)) do od;
%p A140778          b(k), b(abs(a(n-1)-k)):= true$2; k
%p A140778       fi
%p A140778     end:
%p A140778 seq(a(n), n=1..100);  # _Alois P. Heinz_, May 14 2015
%t A140778 a[n_] := a[n] = Module[{}, If [n == 1, b[1] = True; 1, For[k = 1, b[k] || Function[t, b[t] || t == k][Abs[a[n-1] - k]], k++]; {b[k], b[Abs[a[n-1] - k]]} = {True, True}; k]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jan 22 2017, after _Alois P. Heinz_ *)
%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
%o A140778 (PARI) {u=0;a=1;for(n=1,99,u+=1<<a;print1(a",");for(k=1,9e9,(bittest(u,k)||bittest(u,abs(a-k))||a==2*k)&&next;u+=1<<abs(a-k);a=k;break))} \\ _M. F. Hasler_, May 13 2015
%Y A140778 Cf. A140779, A081145.
%Y A140778 See also A005228, A030124, A037257, A037258, A037259, A061577, A257941, A257944.
%K A140778 easy,nonn
%O A140778 1,2
%A A140778 _Franklin T. Adams-Watters_, May 29 2008