A238475 Rectangular array with all start numbers Me(n, k), k >= 1, for the Collatz operation ud^(2*n), n >= 1, ending in an odd number, read by antidiagonals.

1, 9, 5, 17, 37, 21, 25, 69, 149, 85, 33, 101, 277, 597, 341, 41, 133, 405, 1109, 2389, 1365, 49, 165, 533, 1621, 4437, 9557, 5461, 57, 197, 661, 2133, 6485, 17749, 38229, 21845, 65, 229, 789, 2645, 8533, 25941, 70997, 152917, 87381

Offset: 1

Author

Wolfdieter Lang, Mar 10 2014

Keywords

• nonn
• easy
• tabl

Comments

The two operations on natural numbers m used in the Collatz 3x+1 conjecture (see the links) are here (following the M. Trümper reference) denoted 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 positive start numbers Me(n, k), k >= 1, for Collatz sequences following the pattern (word) ud^(2*n), for n >= 1, which end in an odd number. The end number does not depend on n and it is given by Ne(k) = 6*k - 5.

This rectangular array is Example 2.1. with x = 2*n, n >= 1, of the M. Trümper reference, pp. 4-5, written as a triangle by taking NE-SW diagonals. The case x = 2*n+1, n >= 0, for the word ud^(2*k+1) appears as array and triangle in A238476.

The first row sequences of the array Me (they become columns in the triangle Te) are A017077, A238477, A239123, ...

Note that there are also Collatz sequences starting with an odd number, following the pattern ud^(2*n) which end in an even number. For example, take n=1 and the sequence [5, 16, 8, 4]. Such sequences are here not considered.

Examples

			The rectangular array Me(n, k) begins:
n\k      1       2       3        4       5        6        7        8        9       10 ...
1:       1       9      17       25      33       41       49       57       65       73
2:       5      37      69      101     133      165      197      229      261      293
3:      21     149     277      405     533      661      789      917     1045     1173
4:      85     597    1109     1621    2133     2645     3157     3669     4181     4693
5:     341    2389    4437     6485    8533    10581    12629    14677    16725    18773
6:    1365    9557   17749    25941   34133    42325    50517    58709    66901    75093
7:    5461   38229   70997   103765  136533   169301   202069   234837   267605   300373
8:   21845  152917  283989   415061  546133   677205   808277   939349  1070421  1201493
9:   87381  611669 1135957  1660245 2184533  2708821  3233109  3757397  4281685  4805973
10: 349525 2446677 4543829  6640981 8738133 10835285 12932437 15029589 17126741 19223893
...
The triangle Te(m, n) begins (zeros are not shown):
m\n   1    2    3     4      5      6       7       8       9      10 ...
1:    1
2:    9    5
3:   17   37   21
4:   25   69  149    85
5:   33  101  277   597    341
6:   41  133  405  1109   2389   1365
7:   49  165  533  1621   4437   9557    5461
8:   57  197  661  2133   6485  17749   38229   21845
9:   65  229  789  2645   8533  25941   70997  152917   87381
10:  73  261  917  3157  10581  34133  103765  283989  611669  349525
...
----------------------------------------------------------------------------------------------
n=1, ud^2, k=1: Me(1, 1) = 1 = Te(1, 1), Ne(1) = 1 with the Collatz sequence [1, 4, 2, 1] of length 4.
n=1, ud^2, k=2: Me(1, 2) = 9 = Te(2, 1), Ne(2) = 7 with the Collatz sequence [9, 28, 14, 7] of length 4.
n=2, ud^4, k=1: Me(2, 1) = 5 = Te(2, 2), Ne(1) = 1 with the length 6 Collatz sequence [5, 16, 8, 4, 2, 1].
n=5, ud^(10), k=2: Me(5, 2) =  2389  = Te(6,5),  Ne(2) = 7 with the Collatz sequence [2389, 7168, 3584, 1792, 896, 448, 224, 112, 56, 28, 14, 7] of length 12.
		

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, A238476, A017077, A238477, A239123.

Formula

The array: Me(n, k) = 2^(2*n+1)*k - (5*2^(2*n)+1)/3 for n >= 1 and k >= 1.

The triangle: Te(m, n) = Me(n, m-n+1) = 2*4^n*(m-n) + (4^n-1)/3 for m >= n >= 1 and 0 for m < n.