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

0,4

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.

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.

Example

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.

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}]

Crossrefs

Cf. A222118, A359194, A359207, A359218.

Sequence in context: A158215 A294254 A216792 * A260237 A118135 A318671

Adjacent sequences: A359212 A359213 A359214 * A359216 A359217 A359218

Keyword

base,nonn

Author

Michael De Vlieger, Dec 21 2022

Status

approved