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A367646 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - GCT(a(n-2),a(n-1)) if nonnegative and not already in the sequence, else a(n) = a(n-1) + CT(a(n-2),a(n-1)), where CT(a,b) is the Comma transform (cf. A367360) of a and b, while GCT(a,b) is the largest possible generalized Command transform (cf. A367635) where at least one digit of both a and b can be chosen.

%I A367646 #27 Dec 15 2023 09:11:17
%S A367646 0,1,2,14,35,78,21,103,92,53,28,60,146,132,71,44,30,73,66,102,41,17,6,
%T A367646 82,150,129,117,26,98,29,111,20,8,16,97,166,95,164,113,72,109,88,186,
%U A367646 105,166,115,54,109,68,164,83,131,100,89,81,179,62,158,137,56,131,70,87,79,156,65,131,80
%N A367646 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) - GCT(a(n-2),a(n-1)) if nonnegative and not already in the sequence, else a(n) = a(n-1) + CT(a(n-2),a(n-1)), where CT(a,b) is the Comma transform (cf. A367360) of a and b, while GCT(a,b) is the largest possible generalized Command transform (cf. A367635) where at least one digit of both a and b can be chosen.
%C A367646 This is a variation of A367578, where one can choose more than one digit from both a(n-2) and a(n-1) to create the largest possible step to a nonnegative number which has not previously appeared to form a(n). If all such numbers have already appeared the smallest possible forward step is chosen, which is just the standard Comma transform of a(n-2) and a(n-1).
%C A367646 It is conjectured that all nonnegative numbers appear in the sequence. After the first 10 million terms the only fixed points are 0, 1, 2, 29, 65, 84, 222, 377, 491, 499, and it is likely no more exist. The first number to appear twice is a(35) = a(44) = 166.
%H A367646 Scott R. Shannon, <a href="/A367646/b367646.txt">Table of n, a(n) for n = 0..10000</a>
%H A367646 Scott R. Shannon, <a href="/A367646/a367646.png">Image of the first 100000 terms</a>. The green line is a(n) = n.
%H A367646 Scott R. Shannon, <a href="/A367646/a367646_1.png">Image of the first 10 million terms</a>.
%e A367646 a(3) = 14 as CT(a(1),a(2)) = CT(1,2) = 12, so a(3) = a(2) + 12 = 14.
%e A367646 a(6) = 21 as GCT(a(4),a(5)) = GCT(35,78) = 57, so a(6) = a(5) - 57 = 21, as 21 is nonnegative and not already in the sequence.
%e A367646 a(13) = 132 as GCT(a(11),a(12)) = GCT(60,146) = 14, so a(13) = a(12) - 14 = 132, as 132 is nonnegative and not already in the sequence. This is the first term to differ from A367578.
%Y A367646 Cf. A367360, A367635, A367578, A005132, A121805.
%K A367646 nonn,base
%O A367646 0,3
%A A367646 _Scott R. Shannon_, Nov 25 2023