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.

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, 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, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1

Offset: 0

Author

Michael De Vlieger, Dec 21 2022

Keywords

• base
• nonn

Comments

"Branch length" of n->A359194(n).

a(0) = 0 since n = 0.

Let m be the first term in S(n) that has appeared in S(k), k < n. A359218(n) = m.

Analogous to A222118 which instead regards the Collatz function A006318.

Examples

			a(0) = 0 since n = 0.
a(1) = 1 since S(1) = {1, 0}, but m = 0 appeared in S(0).
a(2) = 1 since S(2) = {2, 1, 0}, but m = 1 appeared in S(1).
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).
a(4) = 1 since S(4) = {4, 3, ...} but 3 appears in S(3), etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..16384
• Michael De Vlieger, Log log scatterplot of a(n) and b(n), n = 1..2^14, b(n) = A359207(n) in dark blue, a(n) in red, highlighting where a(n) = b(n) in green.

Crossrefs

Cf. A222118, A359194, A359207, A359218.

Programs

Mathematica

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}]