A235538 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.
1, 3, 9, 26, 5, 13, 31, 15, 27, 81, 22, 45, 92, 20, 50, 145, 46, 89, 32, 71, 151, 40, 75, 163, 73, 124, 60, 126, 244, 97, 219, 63, 132, 306, 68, 144, 297, 79, 166, 354, 83, 187, 394, 94, 203, 419, 108, 220, 460, 127, 260, 110, 247, 513, 161, 340, 117, 252
Offset: 1
Examples
For n=1: - 1 is admissible; hence a(1)=1. For n=2: - 1 is not admissible (as it already appears in the sequence), - 2 is not admissible (as a(1) would appear in the first differences), - 3 is admissible; hence a(2)=3. For n=3: - 1 is not admissible (as it already appears in the sequence), - 2 is not admissible (as it already appears in the first differences), - 3 is not admissible (as it already appears in the sequence), - 4 is not admissible (as a(1) would appear in the first differences), - 5 is not admissible (as 2 would appear twice in the first differences), - 6 is not admissible (as a(2) would appear in the first differences), - 7 is not admissible (as 2 would appear in the first and second differences), - 8 is not admissible (as a(2) would appear in the second differences), - 9 is admissible; hence a(3)=9.
Links
- Paul Tek, Table of n, a(n) for n = 1..1000
- Paul Tek, PERL program for this sequence
Programs
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Mathematica
a[1] = 1; diffs0 = {1} (* flattened array of successive differences *); 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]]] ]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 31 2018 *) -
Perl
See Link section.
Extensions
Added "infinite" to definition. - N. J. A. Sloane, Oct 05 2019