Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #27 Oct 05 2019 18:00:45
%S 1,3,9,26,5,13,31,15,27,81,22,45,92,20,50,145,46,89,32,71,151,40,75,
%T 163,73,124,60,126,244,97,219,63,132,306,68,144,297,79,166,354,83,187,
%U 394,94,203,419,108,220,460,127,260,110,247,513,161,340,117,252
%N Earliest infinite sequence of natural numbers such that the members of this sequence as well as the absolute values of the members of the k-th differences of this sequence, for all k>0, are all distinct.
%H Paul Tek, <a href="/A235538/b235538.txt">Table of n, a(n) for n = 1..1000</a>
%H Paul Tek, <a href="/A235538/a235538.txt">PERL program for this sequence</a>
%e For n=1:
%e - 1 is admissible; hence a(1)=1.
%e For n=2:
%e - 1 is not admissible (as it already appears in the sequence),
%e - 2 is not admissible (as a(1) would appear in the first differences),
%e - 3 is admissible; hence a(2)=3.
%e For n=3:
%e - 1 is not admissible (as it already appears in the sequence),
%e - 2 is not admissible (as it already appears in the first differences),
%e - 3 is not admissible (as it already appears in the sequence),
%e - 4 is not admissible (as a(1) would appear in the first differences),
%e - 5 is not admissible (as 2 would appear twice in the first differences),
%e - 6 is not admissible (as a(2) would appear in the first differences),
%e - 7 is not admissible (as 2 would appear in the first and second differences),
%e - 8 is not admissible (as a(2) would appear in the second differences),
%e - 9 is admissible; hence a(3)=9.
%t a[1] = 1; diffs0 = {1} (* flattened array of successive differences *);
%t a[n_] := a[n] = Module[{}, aa = Array[a, n-1]; m0 = 1; While[ MemberQ[ diffs0, m0], m0++]; For[m = m0, True, m++, am = Append[aa, m]; td = Table[Differences[am, k], {k, 0, n-1}]; diffs = Abs[Flatten[td]]; If[ Length[diffs] == Length[Union[diffs]], diffs0 = diffs//Sort; Return[m]]] ];
%t Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jan 31 2018 *)
%o (Perl) See Link section.
%Y Cf. A005228, A005282, A035312, A235539, A327460, A327887.
%K nonn,nice
%O 1,2
%A _Paul Tek_, Jan 12 2014
%E Added "infinite" to definition. - _N. J. A. Sloane_, Oct 05 2019