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A367467 Lexicographically earliest infinite sequence of positive integers such that a(n + a(n)) is distinct for all n.

%I A367467 #76 Dec 17 2023 10:33:25
%S A367467 1,1,2,2,3,4,2,5,6,7,1,8,9,2,10,11,12,1,13,14,2,15,16,2,17,18,19,2,20,
%T A367467 21,2,22,23,24,1,25,26,2,27,28,2,29,30,31,2,32,33,2,34,35,36,1,37,38,
%U A367467 2,39,40,41,1,42,43,2,44,45,2,46,47,48,1,49,50,2,51,52,53,1,54,55,2,56,57,2,58,59,60,2,61,62,2
%N A367467 Lexicographically earliest infinite sequence of positive integers such that a(n + a(n)) is distinct for all n.
%C A367467 Consider each index i as a location from which one can jump a(i) terms forward. To find a(n) we have to check 2 conditions:
%C A367467 1. The value a(n) can be reached in one jump by at most one previous location.
%C A367467 2. Location n reaches a location in one jump that is not reached in one jump from a location before n.
%C A367467 Described in the above way, the sequence seems to be structured as follows:
%C A367467 A083051 appears to give the indices which cannot be reached from any earlier term; the terms at these indices are 1s and 2s.
%C A367467 A087057 appears to give the indices which can be reached from an earlier term; except for a(2), these terms are first occurrences.
%C A367467 From _Thomas Scheuerle_, Nov 26 2023: (Start)
%C A367467 Empirical observations:
%C A367467 It appears that this sequence consists of the natural numbers in ascending order interspersed by 1 and 2.
%C A367467 If we consider the distance between successive ones, we will observe a nonperiodic pattern: 9,7,17,17,7,10,7,17,7,10,... . It appears that there are only 7, 10 and 17 with the exception of 9 once.
%C A367467 If we consider the distance between successive twos, we will also observe an interesting nonperiodic pattern: 3,7,7,3,4,3,7,3,4,3,7,7,3,... . It appears that this pattern consists only of 3, 4 and 7. (End)
%H A367467 Neal Gersh Tolunsky, <a href="/A367467/b367467.txt">Table of n, a(n) for n = 1..1000</a>
%F A367467 From _Thomas Scheuerle_, Nov 26 2023: (Start)
%F A367467 Conjectures:
%F A367467 a(n) = A049472(n) = floor(n*(1 + 1/sqrt(2))) - n, if n is not in A083051.
%F A367467 a(A083051(n)) = A184119(n+1) - A083051(n).
%F A367467 a(a(A083051(n)) + A083051(n)) + a(A083051(n)) + A083051(n) = A328987(n) = floor((a(A083051(n)) + A083051(n))*(1 + 1/sqrt(2))) = floor(A184119(n+1)*(1 + 1/sqrt(2))). (End)
%e A367467 Initial locations and the (by definition) distinct terms that they reach:
%e A367467      n|  1  2  3  4  5  6  7  8  9
%e A367467   a(n)|  1  1  2  2  3  4  2  5  6
%e A367467           =>1=>2====>3
%e A367467                    ====>4
%e A367467                       =======>5
%e A367467                             ====>6
%e A367467 When we evaluate a(i+a(i)) with each index i, we get a distinct value. When i=1, for example, a(1+a(1))=a(1+1)=a(2)=1;  no other i gives 1 as the solution to a(i+a(i)). When i=4, a(4+a(4))=a(4+2)=a(6)=4, and 4 is likewise a solution unique to i=4.
%o A367467 (MATLAB)
%o A367467 function a = A367467( max_n )
%o A367467     a = [1 1:2*max_n];
%o A367467     for n = 3:max_n
%o A367467         a(n) = 1;
%o A367467         while consistency(a, n) == false
%o A367467             a(n) = a(n)+1;
%o A367467         end
%o A367467     end
%o A367467     a = a(1:max_n);
%o A367467 end
%o A367467 function ok = consistency(a, n)
%o A367467     v = a([1:n] + a(1:n));
%o A367467     ok = (n == length(unique(v)));
%o A367467 end % _Thomas Scheuerle_, Nov 21 2023
%Y A367467 Cf. A367832, A293078, A323420, A359807, A367039, A337226, A366691.
%Y A367467 Cf. A049472, A083051, A087057, A184119, A328987.
%K A367467 nonn
%O A367467 1,3
%A A367467 _Neal Gersh Tolunsky_, Nov 18 2023