A285098 -id:A285098 - cp's OEIS frontend

A070168 Irregular triangle of Terras-modified Collatz problem.

1, 2, 1, 3, 5, 8, 4, 2, 1, 4, 2, 1, 5, 8, 4, 2, 1, 6, 3, 5, 8, 4, 2, 1, 7, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1, 8, 4, 2, 1, 9, 14, 7, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1, 10, 5, 8, 4, 2, 1, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1, 12, 6, 3, 5, 8, 4, 2, 1, 13, 20, 10, 5, 8, 4, 2, 1, 14, 7, 11

Offset: 1

Eric W. Weisstein, Apr 23 2002

The row length of this irregular triangle is A006666(n) + 1 = A064433(n+1), n >= 1. - Wolfdieter Lang, Mar 20 2014

			The irregular triangle begins:
n\k   0   1   2   3   4   5   6   8  9 10  11  12  13  14 ...
1:    1
2:    2   1
3:    3   5   8   4   2   1
4:    4   2   1
5:    5   8   4   2   1
6:    6   3   5   8   4   2   1
7:    7  11  17  26  13  20  10   5  8  4   2   1
8:    8   4   2   1
9:    9  14   7  11  17  26  13  20 10  5   8   4   2   1
10:  10   5   8   4   2   1
11:  11  17  26  13  20  10   5   8  4  2   1
12:  12   6   3   5   8   4   2   1
13:  13  20  10   5   8   4   2   1
14:  14   7  11  17  26  13  20  10  5  8   4   2   1
15:  15  23  35  53  80  40  20  10  5  8   4   2   1
...  formatted by _Wolfdieter Lang_, Mar 20 2014
-------------------------------------------------------------

Reinhard Zumkeller, Rows n = 1..250 of triangle, flattened
J. C. Lagarias, The 3x+1 Problem and its Generalizations, Amer. Math. Monthly 92 (1985) 3-23.
R. Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241-252.
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture

Cf. A070165 (ordinary Collatz case).

Cf. A014682, A248573, A285098 (row sums).

a070168 n k = a070168_tabf !! (n-1) !! (k-1)
a070168_tabf = map a070168_row [1..]
a070168_row n = (takeWhile (/= 1) $ iterate a014682 n) ++ [1]
a070168_list = concat a070168_tabf
-- Reinhard Zumkeller, Oct 03 2014

f[n_] := If[EvenQ[n], n/2, (3 n + 1)/2];
Table[NestWhileList[f, n, # != 1 &], {n, 1, 30}] // Grid (* Geoffrey Critzer, Oct 18 2014 *)

def a(n):
    if n==1: return [1]
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = (3*n + 1)//2
        l.append(n)
        if n<2: break
    return l
for n in range(1, 16): print(a(n)) # Indranil Ghosh, Apr 15 2017

From Wolfdieter Lang, Mar 20 2014: (Start)
See Lagarias, pp. 4-7, eqs. (2.1), (2.4) with (2.5) and (2.6).
T(n,k) = T^{(k)}(n), with the iterations of the Terras-modified Collatz map: T(n) = n/2 if n is even and otherwise (3*n+1)/2, n >= 1. T^{(0)}(n) = n.
T(n,k) = lambda(n,k)*n + rho(n,k), with lambda(n,k) = (3^X(n,k,-1))/2^k and rho(n,k) = sum(x(n,j)*(3^X(n,k,j))/ 2^(k-j), j=0..(k-1)) with X(n,k,j) = sum(x(n,j+p), p=1.. (k-1-j)) where x(n,j) = T^{(j)}(n) (mod 2). The parity sequence suffices to determine T(n,k).
(End)

Name shortened, tabl changed into tabf, Cf. added by Wolfdieter Lang, Mar 20 2014

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.

Original entry on oeis.org

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

Paolo Xausa, Feb 22 2022

nonn

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.

			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.

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.

Cf. A006666, A070168, A285098, A351849.

(* 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}]

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)])

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

A375268 Row sums of A375266.

Original entry on oeis.org

1, 3, 4, 7, 36, 9, 288, 15, 13, 46, 259, 19, 119, 302, 51, 31, 214, 27, 519, 66, 309, 281, 633, 39, 658, 145, 40, 330, 442, 76, 101104, 63, 292, 248, 540, 55, 535, 557, 158, 106, 101331, 344, 1338, 325, 96, 679, 100979, 79, 806, 708, 265, 197, 399, 81, 102316, 386

Offset: 1

Paolo Xausa, Aug 09 2024

Cf. A375265, A375266, A375267, A375280.

Cf. A033493, A285098.

A375265[n_] := Which[Divisible[n, 3], n/3, Divisible[n, 2], n/2, True, 3*n + 1];
Array[Total[NestWhileList[A375265, #, # > 1 &]] &, 100]

a(n) = Sum_{k = 1..A375267(n) + 1} A375266(n,k).

A375910 Row sums of A350279.

Original entry on oeis.org

1, 4, 9, 48, 13, 41, 61, 24, 30, 72, 69, 151, 86, 40, 53, 538, 74, 128, 109, 100, 110, 182, 69, 507, 135, 81, 93, 395, 129, 217, 599, 132, 139, 249, 220, 460, 182, 161, 177, 850, 121, 340, 267, 140, 158, 448, 631, 1625, 232, 173, 182, 708, 233, 389, 504, 220, 242, 428

Offset: 1

Paolo Xausa, Sep 02 2024

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

Cf. A349407, A350279, A375909, A375911.

Cf. A033493, A285098.

FarkasStep[x_] := Which[Divisible[x, 3], x/3, Mod[x, 4] == 3, (3*x + 1)/2, True, (x + 1)/2];
Array[Total[FixedPointList[FarkasStep, 2*# - 1]] - 1 &, 100]

a(n) = Sum_{k = 1..A375909(n) + 1} A350279(n,k).

cp's OEIS Frontend

A070168 Irregular triangle of Terras-modified Collatz problem.

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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.

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A375268 Row sums of A375266.

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A375910 Row sums of A350279.

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