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

1,2

Comments

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

Links

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

Formula

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)

Example

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
-------------------------------------------------------------

Mathematica

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 *)

Prog

(Haskell)
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
(Python)
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

Crossrefs

Cf. A070165 (ordinary Collatz case).

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

Sequence in context: A319153 A286390 A135017 * A246646 A198094 A263047

Adjacent sequences: A070165 A070166 A070167 * A070169 A070170 A070171

Keyword

nonn,easy,tabf

Author

Eric W. Weisstein, Apr 23 2002

Extensions

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

Status

approved