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 A381003 #62 Mar 03 2025 13:26:36
%S A381003 0,1,1,2,1,3,1,4,4,1,5,6,5,1,6,8,1,9,1,10,10,1,11,12,1,13,13,1,14,11,
%T A381003 15,1,16,17,1,18,5,7,18,14,12,6,20,3,22,23,15,24,17,1,18,3,26,27,20,2,
%U A381003 28,29,1,30,4,16,31,1,32,18,33,10,34,1,35,36,36,2,21,38
%N A381003 Lexicographically earliest sequence with a(0) = 0 and a(n) = a(n + a(n)) - a(n - a(n)) > 0.
%C A381003 A trivial but not lexicographically earliest solution for this problem is given by a(m) = ceiling(m/2) (A110654).
%C A381003 We assume a(<0) = 0 in this definition but conjecture that a(n) < n for n > 1.
%C A381003 If we iterate recursively k_{n+1} = a(k_{n})+k_{n} starting with k{0}=2, then we will observe a monotone increasing trajectory. An interesting consequence of the existence of this sequence is that if we start a trajectory at some k{m} > 2, such a trajectory would in the majority of cases ( or in all such cases? ) merge into the trajectory which is starting at 2.
%H A381003 Thomas Scheuerle, <a href="/A381003/b381003.txt">Table of n, a(n) for n = 0..10000</a>
%H A381003 Thomas Scheuerle, <a href="/A381003/a381003.png">Graph of Sum_{k=0..n} a(k)/n over n</a>, it looks almost like a straight line with slope 0.14.. . Is there any relation to a well known constant?
%H A381003 Thomas Scheuerle, <a href="/A381003/a381003_1.png">Scatter plot of the first 15000 values</a>, colored by first root ancestor the colormap has period 500.
%H A381003 Thomas Scheuerle, <a href="/A381003/a381003_2.png">Scatter plot of the first 15000 values</a>, colored by second root ancestor the colormap has period 500.
%H A381003 Thomas Scheuerle, <a href="/A381003/a381003_3.png">Scatter plot of the first 15000 values</a>, colored by length of the ancestor chain for the first ancestors the colormap has period 15.
%H A381003 Thomas Scheuerle, <a href="/A381003/a381003_4.png">Scatter plot of the first 15000 values</a>, colored by length of the ancestor chain for the second ancestors the colormap has period 5.
%F A381003 a(a(n) + n) >= a(n).
%e A381003 a(10) = 5 = a(10 + 5) - a(10 - 5) = 8 - 3.
%Y A381003 Cf. A110654.
%K A381003 nonn
%O A381003 0,4
%A A381003 _Thomas Scheuerle_, Feb 11 2025