A240223 Rectangular companion array to M(n,k), given in A240222, showing the end numbers N(n, k), k >= 1, for the Collatz operation (udd)^(n-1) ud, n >= 1, read by antidiagonals.

2, 5, 2, 8, 11, 2, 11, 20, 29, 2, 14, 29, 56, 83, 2, 17, 38, 83, 164, 245, 2, 20, 47, 110, 245, 488, 731, 2, 23, 56, 137, 326, 731, 1460, 2189, 2, 26, 65, 164, 407, 974, 2189, 4376, 6563, 2, 29, 74, 191, 488, 1217, 2918, 6563, 13124, 19685, 2, 32, 83, 218, 569, 1460, 3647, 8750, 19685, 39368, 59051, 2

Offset

0,1

Comments

The companion array and triangle for the start numbers M(n, k) is given in A240222.

For the Collatz operations u (for 'up') and d (for 'down') see the comment on A240222, also for links, especially for the M. Trümper paper.

Links

Table of n, a(n) for n=0..65.

Wolfdieter Lang, Rectangular array and triangle.

Wolfdieter Lang, On Collatz Words, Sequences, and Trees, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.

Manfred Trümper, The Collatz Problem in the Light of an Infinite Free Semigroup, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.

Formula

The array: N(n, k) = 2 + 3^n*k for n >= 1 and k >= 0.

The triangle: TN(m, n) = N(n,m-n+1) = 2 + 3^n*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n.

Example

The rectangular array N(n, k) begins
  n\k 0      1       2       3       4       5 ...
  1:  2      5       8      11      14      17
  2:  2     11      20      29      38      47
  3:  2     29      56      83     110     137
  4:  2     83     164     245     326     407
  5:  2    245     488     731     974    1217
  6:  2    731    1460    2189    2918    3647
  7:  2   2189    4376    6563    8750   10937
  8:  2   6563   13124   19685   26246   32807
  9:  2  19685   39368   59051   78734   98417
  10: 2  59051  118100  177149  236198  295247
  ...
For more columns see the link.
The triangle TN(m, n) begins (zeros are not shown):
  m\n  1  2   3   4    5    6    7 ...
  0:   2
  1:   5  2
  2:   8 11   2
  3:  11 20  29   2
  4:  14 29  56  83    2
  5:  17 38  83 164  245    2
  6:  20 47 110 245  488  731    2
  ...
For more rows see the link.
n=1, ud, k=0: M(1, 0) = 1, N(1, 0) = TN(0, 1) = 2 with the Collatz sequence [1, 4, 2] of length 3.
n=1, ud, k=2: M(1, 2) = 5, N(1, 2) = TN(2, 1) = 8 with the Collatz sequence [5, 16, 8] of length 3.
n=2, uddud, k=0: M(2, 0) = 1, Ne(2, 0) = TN(1, 2) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9.

Crossrefs

Cf. A238475, A238476, A239126, A239127, A240222, A016789 (first row of N), A017185 (second row of N).

Sequence in context: A274415 A292584 A029621 * A134349 A354709 A131711

Adjacent sequences: A240220 A240221 A240222 * A240224 A240225 A240226

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Apr 04 2014

Status

approved