This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354223 #42 Jul 18 2022 10:00:38
%S A354223 1,2,3,1,3,4,2,1,2,4,1,4,5,3,2,1,2,3,1,3,5,2,1,2,5,1,5,6,4,3,2,1,2,3,
%T A354223 1,3,4,2,1,2,4,1,4,6,3,2,1,2,3,1,3,6,2,1,2,6,1,6,7,5,4,3,2,1,2,3,1,3,
%U A354223 4,2,1,2,4,1,4,5,3,2,1,2,3,1,3,5,2,1,2,5,1,5,7,4,3,2,1,2,3,1,3,4,2,1,2,4,1,4,7,3,2,1,2,3,1,3,7,2,1,2,7,1,7
%N A354223 The initial sequence of the positive integers gradually becomes interspersed with the copies of its terms and so is transformed into this sequence. The method is described in the Comments section.
%C A354223 Pre-inked rubberstamps of the positive integers 1, 2, 3, ... are lined up on an infinite paper strip, with sufficient gaps between them. Underneath them already there are their imprints on the paper. We lift and move the first stamp over as many others as is the number of the second one in the line. Then, we press it down on the paper in the gap between the two stamps at that location. We leave it to stand there on its new imprint until its next turn. We successively repeat this with every temporary leading pair: the second stamp will govern the first one, how many others it has to leap over. The ink-printed numbers left uncovered on the paper in front of the receding line of the stamps form the sequence.
%C A354223 It appears that for every new term k > 1, runs of 2^(k-1)-1 length are in this sequence, with k being at the beginning and at the end of these runs. Also, within these runs k repeats itself in the self-same manner, only in reverse order. Could this indicate a fractal property?
%C A354223 If you take the first differences of this sequence as a new sequence and remove all negative numbers from it, then you will obtain A089309. - _Thomas Scheuerle_, May 20 2022
%H A354223 Thomas Scheuerle, <a href="/A354223/b354223.txt">Table of n, a(n) for n = 1..6000</a>
%H A354223 Thomas Scheuerle, <a href="/A354223/a354223.png">First differences of a(n) as scatter plot from 1 to 10000.</a>
%F A354223 From _Thomas Scheuerle_, May 19 2022: (Start)
%F A354223 a(2^(m+1) + n - 1) - a(n) < 2, if n < 2^(m+1) - m. This evaluates to zero for all n with m+1 exceptions. For these exceptions it will become 1. The exceptions are n = {A132045(m), A132045(m)+2^m, A132045(m)+2^m+2^(m-1), A132045(m)+2^m+2^(m-1)+2^(m-2), ..., A132045(m)+2^(m+1)-1}.
%F A354223 (1/n)*Sum_{k=1..n} a(k) <= 3. It appears that this arithmetic mean converges to 3 if n approaches infinity.
%F A354223 a(1 + Sum_{k=1..n} A090739(k)) = a(1 + A120738(n - 1)) = 1.
%F A354223 a(Sum_{k=1..n} (1 + A195986(k))) = 2. (End)
%e A354223 The initial line of the stamps:
%e A354223 (1) (2) (3) (4) (5) ...
%e A354223 Stamp (1) leaps over two others as is governed by stamp (2), leaving its imprint 1 in front:
%e A354223 1 (2) (3) (1) (4) (5) ...
%e A354223 Stamp (2) leaps over three others as stamp (3) dictates. Its imprint 2 remains on the paper:
%e A354223 1 2 (3) (1) (4) (2) (5) ...
%e A354223 Stamp (3) leaps over only the stamp (1), leaving its imprint 3 behind:
%e A354223 1 2 3 (1) (3) (4) (2) (5) ...
%e A354223 Now stamp (1) leaps again, over three others into the gap between stamps (2) and (5). Its print 1 remains:
%e A354223 1 2 3 1 (3) (4) (2) (1) (5) ...
%e A354223 (And so on.)
%o A354223 (MATLAB)
%o A354223 function a = A354223( max_n )
%o A354223 a = [1:max_n];
%o A354223 for n = 1:max_n
%o A354223 m = a(n);
%o A354223 j = a(n+1);
%o A354223 a = [a(1:n+j) m a(n+j+1:end)];
%o A354223 end
%o A354223 a = a(1:max_n);
%o A354223 end % _Thomas Scheuerle_, May 20 2022
%Y A354223 Cf. A000027, A089309, A090739, A120738, A132045, A195986.
%K A354223 nonn
%O A354223 1,2
%A A354223 _Tamas Sandor Nagy_, May 19 2022