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A359215 Number of terms in S(n) that did not appear in previous trajectories, where S(n) is the trajectory of the mappings of x->A359194(x) starting with n and stopping when 0 is reached, -1 if 0 is never reached.

%I A359215 #17 Dec 23 2022 11:22:28
%S A359215 0,1,1,11,1,1,0,2,1,1,0,6,78,0,2,0,0,1,0,1,1,0,0,3,0,0,11,0,7571,2,0,
%T A359215 0,1,1,0,1,1,0,1,0,0,1,1,0,1,3,0,3,77,0,5419,1,0,1,4,0,1,0,0,2,2,0,2,
%U A359215 0,0,1,0,1,1,0,1,1,0,1,1,0,0,1,0,1,1,0,1,1
%N A359215 Number of terms in S(n) that did not appear in previous trajectories, where S(n) is the trajectory of the mappings of x->A359194(x) starting with n and stopping when 0 is reached, -1 if 0 is never reached.
%C A359215 "Branch length" of n->A359194(n).
%C A359215 a(0) = 0 since n = 0.
%C A359215 Let m be the first term in S(n) that has appeared in S(k), k < n. A359218(n) = m.
%C A359215 Analogous to A222118 which instead regards the Collatz function A006318.
%H A359215 Michael De Vlieger, <a href="/A359215/b359215.txt">Table of n, a(n) for n = 0..16384</a>
%H A359215 Michael De Vlieger, <a href="/A359215/a359215.png">Log log scatterplot of a(n) and b(n)</a>, n = 1..2^14, b(n) = A359207(n) in dark blue, a(n) in red, highlighting where a(n) = b(n) in green.
%e A359215 a(0) = 0 since n = 0.
%e A359215 a(1) = 1 since S(1) = {1, 0}, but m = 0 appeared in S(0).
%e A359215 a(2) = 1 since S(2) = {2, 1, 0}, but m = 1 appeared in S(1).
%e A359215 a(3) = 11 since S(3) = {3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0}, but m = 0 appeared in S(0).
%e A359215 a(4) = 1 since S(4) = {4, 3, ...} but 3 appears in S(3), etc.
%t A359215 c[_] = -1; c[0] = 0; f[n_] := FromDigits[BitXor[1, IntegerDigits[3*n, 2]], 2]; Table[(Map[If[c[#1] == -1, Set[c[#1], #2]] & @@ # &, Partition[#, 2, 1]]; -1 + Length[#]) &@ NestWhileList[f, n, c[#] == -1 &], {n, 0, 120}]
%Y A359215 Cf. A222118, A359194, A359207, A359218.
%K A359215 base,nonn
%O A359215 0,4
%A A359215 _Michael De Vlieger_, Dec 21 2022