A239127 Rectangular companion array to M(n,k), given in A239126, showing the end numbers N(n, k), k >= 1, for the Collatz operation (ud)^n, n >= 1, ending in an odd number, read by antidiagonals.

5, 11, 17, 17, 35, 53, 23, 53, 107, 161, 29, 71, 161, 323, 485, 35, 89, 215, 485, 971, 1457, 41, 107, 269, 647, 1457, 2915, 4373, 47, 125, 323, 809, 1943, 4373, 8747, 13121, 53, 143, 377, 971, 2429, 5831, 13121, 26243, 39365, 59, 161, 431, 1133, 2915, 7289, 17495, 39365, 78731, 118097

Offset: 1

Author

Wolfdieter Lang, Mar 13 2014

Keywords

• nonn
• tabl
• easy

Comments

The companion array and triangle for the odd start numbers M(n, k) is given in A239126.

See the comments on A239126 for the Collatz 3x+1 problem and the u and d operations.

This rectangular array is N of the Example 2.2. with x=y = n, n >= 1, of the M. Trümper reference, pp. 7-8, written as a triangle by taking NE-SW diagonals. The Collatz sequence starting with odd M(n, k) from A239126 and ending in odd N(n, k) has length 2*n+1 for each k.

The first row sequences of the array N (columns of triangle TN) are A016969, A239129, ...

Examples

			The rectangular array N(n, k) begins:
n\k      1      2      3      4      5      6     7       8       9      10 ...
1:       5     11     17     23     29     35     41     47      53      59
2:      17     35     53     71     89    107    125    143     161     179
3:      53    107    161    215    269    323    377    431     485     539
4:     161    323    485    647    809    971   1133   1295    1457    1619
5:     485    971   1457   1943   2429   2915   3401   3887    4373    4859
6:    1457   2915   4373   5831   7289   8747  10205  11663   13121   14579
7:    4373   8747  13121  17495  21869  26243  30617  34991   39365   43739
8:   13121  26243  39365  52487  65609  78731  91853 104975  118097  131219
9:   39365  78731 118097 157463 196829 236195 275561 314927  354293  393659
10: 118097 236195 354293 472391 590489 708587 826685 944783 1062881 1180979
...
-------------------------------------------------------------------------------
The triangle TN(m, n) begins (zeros are not shown):
m\n   1   2  3     4    5    6     7     8     9     10 ...
1:    5
2:   11  17
3:   17  35  53
4:   23  53 107  161
5:   29  71 161  323  485
6:   35  89 215  485  971 1457
7:   41 107 269  647 1457 2915  4373
8:   47 125 323  809 1943 4373  8747 13121
9:   53 143 377  971 2429 5831 13121 26243 39365
10:  59 161 431 1133 2915 7289 17495 39365 78731 118097
...
n=1, ud, k=1: M(1, 1) = 3 = TM(1, 1), N(1,1) = 5 with the Collatz sequence  [3, 10, 5] of length 3.
n=1, ud, k=2: M(1, 2) = 7 = TM(2, 1), N(1,2) = 11 with the Collatz sequence  [7, 22, 11] of length 3.
n=4, (ud)^4, k=2: M(4, 2) = 63 = TM(5, 4), N(4,2) = 323 with the Collatz sequence  [63, 190, 95, 286, 143, 430, 215, 646, 323] of length 9.
n=5, (ud)^5, k=1: M(5, 1) = 63 =  TM(5, 5), N(5,1) = 485 with the Collatz sequence  [63, 190, 95, 286, 143, 430, 215, 646, 323, 970, 485]  of length 11.
		

Links

• 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.

Crossrefs

Cf. A006577, A139399, A112695, A239126, A016969, A239129.

Formula

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

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