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A362373 a(0) = 0; for n > 0, if n appears in the sequence then a(n) is the sum of the indices of all previous appearances of n. Otherwise a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

%I A362373 #49 Jun 21 2025 00:45:02
%S A362373 0,1,3,2,6,11,4,11,19,10,9,12,11,24,38,23,7,24,42,8,28,49,27,15,30,5,
%T A362373 31,22,20,49,24,26,58,25,59,94,130,93,14,53,13,54,18,61,17,62,16,63,
%U A362373 111,50,49,100,48,39,41,96,40,97,32,34,94
%N A362373 a(0) = 0; for n > 0, if n appears in the sequence then a(n) is the sum of the indices of all previous appearances of n. Otherwise a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.
%C A362373 Conjecture: All nonnegative integers appear in this sequence. - _Yifan Xie_, Apr 24 2023
%H A362373 Michael De Vlieger, <a href="/A362373/b362373.txt">Table of n, a(n) for n = 0..16384</a>
%H A362373 Michael De Vlieger, <a href="/A362373/a362373.png">Log log scatterplot of a(n)</a>, n = 1..2^14.
%H A362373 Michael De Vlieger, <a href="/A362373/a362373_1.png">Log log scatterplot of a(n)</a>, n = 1..2^20
%e A362373 a(2) = 3 = n, thus a(3) = 2.
%e A362373 a(5) = 11, as 5 has not previously appeared in the sequence, but a(4) - 5 = 1 has, thus a(5) = a(4) + 5 = 6 + 5 = 11.
%e A362373 a(5) and a(7) = 11, and 5 + 7 = 12, thus a(11) = 12.
%t A362373 nn = 120; c[_] := 0; j = a[0] = 0; Do[If[# > 0, Set[k, #], If[And[n <= j, c[#] == 0], Set[k, #], Set[k, j + n]] &[j - n] ] &[c[n]]; c[k] += n; Set[{a[n], j}, {k, k}], {n, nn}]; Array[a, nn, 0] (* _Michael De Vlieger_, Apr 19 2023 *)
%o A362373 (Python)
%o A362373 from itertools import count, islice
%o A362373 def A362373_gen(): # generator of terms
%o A362373     a, ndict = 0, {0:0}
%o A362373     yield 0
%o A362373     for n in count(1):
%o A362373         yield (a:= ndict[n] if n in ndict else (a-n if a>=n and a-n not in ndict else a+n))
%o A362373         ndict[a] = ndict.get(a,0)+n
%o A362373 A363373_list = list(islice(A362373_gen(),30)) # _Chai Wah Wu_, Jun 29 2023
%Y A362373 Cf. A005132, A340612.
%K A362373 nonn,look,easy
%O A362373 0,3
%A A362373 _Kelvin Voskuijl_, Apr 17 2023