A238476 Rectangular array with all start numbers Mo(n, k), k >= 1, for the Collatz operation ud^(2*n-1), n >= 1, ending in an odd number, read by antidiagonals.

3, 7, 13, 11, 29, 53, 15, 45, 117, 213, 19, 61, 181, 469, 853, 23, 77, 245, 725, 1877, 3413, 27, 93, 309, 981, 2901, 7509, 13653, 31, 109, 373, 1237, 3925, 11605, 30037, 54613, 35, 125, 437, 1493, 4949, 15701, 46421, 120149, 218453

Offset: 1

Author

Wolfdieter Lang, Mar 10 2014

Keywords

• nonn
• tabl
• easy

Comments

The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here denoted (with M. Trümper, see the link) by u for 'up' and d for 'down': u m = 3*m+1, if m is odd, and d m = m/2 if m is even. The present array gives all start numbers Mo(n, k), k >= 1, for Collatz sequences following the pattern (word) ud^(2*n-1), with n >= 1, ending in an odd number. This end number does not depend on n and it is given by No(k) = 6*k - 1. This Collatz sequence has length 1 + (1 + 2*n - 1) = 2*n + 1.

This rectangular array is Example 2.1. with x = 2*n-1, n >= 1, of the M. Trümper reference, pp. 4-5, written as a triangle by taking NE-SW diagonals. The case x = 2*n, n >= 1, for the word ud^(2*n) appears as array and triangle A238475.

The first rows of array Mo (columns of triangle To) are A004767, A082285, A239124, ...

Examples

			The rectangular array Mo(n, k) begins:
n\k      1        2        3        4        5        6        7        8        9        10 ...
1:       3        7       11       15       19       23       27       31       35        39
2:      13       29       45       61       77       93      109      125      141       157
3:      53      117      181      245      309      373      437      501      565       629
4:     213      469      725      981     1237     1493     1749     2005     2261      2517
5:     853     1877     2901     3925     4949     5973     6997     8021     9045     10069
6:    3413     7509    11605    15701    19797    23893    27989    32085    36181     40277
7:   13653    30037    46421    62805    79189    95573   111957   128341   144725    161109
8:   54613   120149   185685   251221   316757   382293   447829   513365   578901    644437
9:  218453   480597   742741  1004885  1267029  1529173  1791317  2053461  2315605   2577749
10: 873813  1922389  2970965  4019541  5068117  6116693  7165269  8213845  9262421  10310997
...
---------------------------------------------------------------------------------------------
The triangle To(m, n) begins (zeros are not shown):
m\n    1    2    3     4     5      6      7       8       9      10 ...
1:     3
2:     7   13
3:    11   29   53
4:    15   45  117   213
5:    19   61  181   469   853
6:    23   77  245   725  1877   3413
7:    27   93  309   981  2901   7509  13653
8:    31  109  373  1237  3925  11605  30037   54613
9:    35  125  437  1493  4949  15701  46421  120149  218453
10:   39  141  501  1749  5973  19797  62805  185685  480597  873813
...
n=1, ud, k=1: Mo(1, 1) = 3 = To(1, 1), No(1) = 5 with the Collatz sequence [3, 10, 5] of length 3.
n=1, ud, k=2: Mo(1, 2) = 7 = Te(2, 1), No(2) = 11 with the Collatz sequence [7, 22, 11] of length 3.
n=5, ud^9, k=2: Mo(5, 2) = 1877 = Te(6,5), No(2) = 11 with the Collatz sequence [1877, 5632, 2816, 1408, 704, 352, 176, 88, 44, 22, 11] of length 11.
		

Links

• W. 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.
• Eric Weisstein's World of Mathematics, Collatz Problem.
• Wikipedia, Collatz Conjecture.

Crossrefs

Cf. A006577, A139399, A112695, A238475, A004767, A082285, A239124.

Formula

Mo(n, k) = 2^(2*n)*k - (2^(2*n-1)+1)/3 for n >= 1 and k >= 1.

To(m, n) = Mo(n, m-n+1) = 2^(2*n)*(m-n+1) - (2^(2*n-1)+1)/3 for m >= n >= 1 and 0 for m < n.