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A367849 Lexicographically earliest infinite sequence of positive integers such that for each n, the values in a path of locations starting from any i=n are all distinct, where jumps are allowed from location i to i+a(i).

%I A367849 #45 Dec 17 2023 02:32:42
%S A367849 1,1,2,1,3,1,1,4,1,1,2,5,3,1,1,4,6,1,1,5,1,1,7,1,6,1,1,2,1,8,7,1,1,2,
%T A367849 1,3,1,9,4,1,1,2,5,3,1,1,10,6,1,1,2,1,3,7,1,4,11,1,1,5,8,1,1,2,6,3,1,
%U A367849 12,9,1,7,1,1,2,1,3,1,10,4,13,1,1,5,1,1,2,1,11,1,1,2,1,14,1,1,2,1,3
%N A367849 Lexicographically earliest infinite sequence of positive integers such that for each n, the values in a path of locations starting from any i=n are all distinct, where jumps are allowed from location i to i+a(i).
%C A367849 Consider each index i as a location from which one can jump a(i) terms forward. No starting index can reach the same value more than once by forward jumps.
%C A367849 The value a(i) at the starting index is not part of the path (and thus allows a(2)=1).
%C A367849 The indices of first occurrences are given by A133263 (essentially triangular numbers + 2).
%C A367849 Changing the definition so that jumps are allowed only from location i to i-a(i) gives A002260.
%H A367849 Neal Gersh Tolunsky, <a href="/A367849/b367849.txt">Table of n, a(n) for n = 1..10000</a>
%H A367849 Thomas Scheuerle, <a href="/A367849/a367849.png">Plot of a(1..100000)^2</a>. It is conjectured that only the topmost straight line in this plot will extend into infinity.
%H A367849 Thomas Scheuerle, <a href="/A367849/a367849_1.png">Partial view of a(1..80000)^2 equalized to Y axis by shear mapping</a>. This shows the different lengths of the increasing sequences.
%H A367849 Neal Gersh Tolunsky, <a href="/A367849/a367849_2.png">Ordinal transform of 20000 terms</a>
%H A367849 Neal Gersh Tolunsky, <a href="/A367849/a367849_3.png">First differences of 20000 terms</a>
%F A367849 a(A133263(n)) = n + 1.
%e A367849 We can see, for example, that the terms reachable by jumping forward continuously from i=1 are all distinct (and in this case are just the positive integers):
%e A367849   1, 1, 2, 1, 3, 1, 1, 4, 1, 1, 2, 5
%e A367849   *->1->2---->3------->4---------->5
%e A367849 Beginning at i=9 and jumping forward continuously, we get the sequence 1,2,3,4,5,6,7,9 (in which all terms are likewise distinct).
%o A367849 (MATLAB)
%o A367849 function a = A367849( max_n )
%o A367849     a = zeros(1,max_n); j = find(a == 0,1);
%o A367849     while ~isempty(j)
%o A367849         a(j) = 1; k = 1;
%o A367849         if j+k < max_n
%o A367849             while a(j+k) == 0
%o A367849                 a(j+k) = k;
%o A367849                 if j+2*k-1 < max_n
%o A367849                     j = j+(k-1); k = k+1;
%o A367849                 else
%o A367849                     break;
%o A367849                 end
%o A367849             end
%o A367849         end
%o A367849         j = find(a == 0,1);
%o A367849     end
%o A367849 end % _Thomas Scheuerle_, Dec 09 2023
%Y A367849 Cf. A367467, A367832, A133263 (index of first occurrences), A362248.
%K A367849 nonn
%O A367849 1,3
%A A367849 _Neal Gersh Tolunsky_, Dec 08 2023