A351850 a(n) is the number of iterations of the computation of the A351849 tag system when started from the word encoding n, or -1 if the number of iterations is infinite.

0, 2, 24, 6, 20, 30, 128, 14, 152, 30, 120, 42, 64, 142, 300, 30, 108, 170, 236, 50, 84, 142, 284, 66, 300, 90, 40656, 170, 216, 330, 40524, 62, 384, 142, 260, 206, 264, 274, 996, 90, 40628, 126, 596, 186, 256, 330, 40492, 114, 388, 350, 520, 142, 224, 40710

Offset

1,2

Comments

If x is even, the A351849 tag system evolves from the word encoding x to the word encoding x/2 in x iterations; if x is odd, x+1 iterations are required to produce the word encoding (3x+1)/2.

See A351849 for additional comments, links and examples.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Liesbeth De Mol, Tag systems and Collatz-like functions, Theoretical Computer Science, Volume 390, Issue 1, 2008, pp. 92-101.

Index entries for sequences related to 3x+1 (or Collatz) problem.

Formula

a(n) = (Sum_{k=0..A006666(n)} 2*floor((A070168(n,k)+1)/2)) - 2.

Example

When started from 1111 (the word encoding the number 4), the system evolves as 1111 -> 1123 -> 2323 -> 231 -> 11 -> 23 -> 1, reaching the word 1 after 6 steps. a(4) is therefore 6.

Mathematica

(* First program, based on the tag system definition *)
t[s_]:=StringDrop[s, 2]<>StringReplace[StringTake[s, 1], {"1"->"23", "2"->"1", "3"->"111"}];
nterms=100; Table[Length[NestWhileList[t, StringRepeat["1", n], #!="1"&]]-1, {n, nterms}]
(* Second program, more efficient, based on formula *)
c[x_]:=If[OddQ[x], (3x+1)/2, x/2];
nterms=100; Table[Total[Map[If[OddQ[#], #+1, #]&, NestWhileList[c, n, #>1&]]]-2, {n, nterms}]

Prog

(Python)
def A351850(n):
    s, steps = "1" * n, 0
    while s != "1":
        if s[0] == "1": s += "23"
        elif s[0] == "2": s += "1"
        else: s += "111"
        s = s[2:]
        steps += 1
    return steps
nterms = 100
print([A351850(n) for n in range(1, nterms + 1)])

Crossrefs

Cf. A006666, A070168, A285098, A351849.

Sequence in context: A079612 A329263 A227477 * A066585 A278563 A075267

Adjacent sequences: A351847 A351848 A351849 * A351851 A351852 A351853

Keyword

nonn

Author

Paolo Xausa, Feb 22 2022

Status

approved