A257941 Lexicographically earliest sequence of positive integers such that the terms and their absolute first differences are all distinct and no term is the sum of two distinct earlier terms.

1, 3, 7, 12, 18, 26, 9, 20, 34, 24, 39, 55, 22, 45, 66, 28, 47, 72, 85, 49, 76, 108, 68, 99, 53, 82, 112, 70, 114, 149, 74, 122, 172, 93, 145, 203, 101, 160, 95, 162, 216, 118, 187, 224, 141, 214, 143, 235, 139, 195, 281, 164, 241, 329, 166, 260, 170, 283, 168

Offset: 1

Author

Eric Angelini and Alois P. Heinz, May 13 2015

Keywords

• nonn
• look

Comments

The sequence of absolute first differences begins: 2, 4, 5, 6, 8, 17, 11, 14, 10, 15, 16, 33, 23, 21, 38, 19, 25, 13, 36, 27, 32, 40, ... .

The sequence is 0-additive.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• E. Angelini et al., 0-additive and first differences and follow-up messages on the SeqFan list, May 13 2015
• Eric Weisstein's World of Mathematics, s-Additive Sequence

Crossrefs

Cf. A005228, A030124, A033627, A095115, A140778, A257944.

Programs

Maple

s:= proc() false end: b:= proc() false end:
a:= proc(n) option remember; local i, k;
      if n=1 then b(1):= true; 1
    else for k while b(k) or s(k) or
         (t-> b(t) or t=k)(abs(a(n-1)-k)) do od;
         for i to n-1 do s(a(i)+k):= true od;
         b(k), b(abs(a(n-1)-k)):= true$2; k
      fi
    end:
seq(a(n), n=1..101);

Mathematica

s[] = False; b[] = False;
a[n_] := a[n] = Module[{i, k}, If[n == 1, b[1] = True; 1, For[k = 1, b[k] || s[k] || Function[t, b[t] || t == k][Abs[a[n-1]-k]], k++]; For[i = 1, i <= n-1, i++, s[a[i]+k] = True]; {b[k], b[Abs[a[n-1]-k]]} = {True, True}; k]];
Array[a, 101] (* Jean-François Alcover, Oct 28 2020, after Maple *)