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.

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, 105, 37, 77, 118, 42, 88, 135, 48, 97, 147, 52, 103, 156, 56, 113, 171, 59, 120, 184, 65, 131, 198, 71, 143, 216, 74, 149, 227, 79, 159, 240, 82, 165, 249, 86, 175, 265, 91, 183

Offset: 1

Author

Franklin T. Adams-Watters, May 29 2008

Keywords

• easy
• nonn

Comments

This sequence and its first differences include every positive integer (exactly once).

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A140779, A081145.

See also A005228, A030124, A037257, A037258, A037259, A061577, A257941, A257944.

Programs

Maple

b:= proc() false end:
a:= proc(n) option remember; local k;
      if n=1 then b(1):= true; 1
    else for k while b(k) or (t-> b(t) or t=k)(abs(a(n-1)-k)) do od;
         b(k), b(abs(a(n-1)-k)):= true$2; k
      fi
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 14 2015

Mathematica

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 *)

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

PARI

{u=0;a=1;for(n=1,99,u+=1<M. F. Hasler, May 13 2015