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%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
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