A240222 Rectangular array giving all start values M(n, k), k >= 1, for Collatz sequences following the pattern (udd)^(n-1) ud, n >= 1, read by antidiagonals.

1, 3, 1, 5, 9, 1, 7, 17, 33, 1, 9, 25, 65, 129, 1, 11, 33, 97, 257, 513, 1, 13, 41, 129, 385, 1025, 2049, 1, 15, 49, 161, 513, 1537, 4097, 8193, 1, 17, 57, 193, 641, 2049, 6145, 16385, 32769, 1, 19, 65, 225, 769, 2561, 8193, 24577, 65537, 131073, 1, 21, 73, 257, 897, 3073, 10241, 32769, 98305

Offset: 1

Author

Wolfdieter Lang, Apr 02 2014

Keywords

• nonn
• easy
• tabl

Comments

The companion array and triangle for the end numbers N(n, k) is given in A240223.

The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here (following the M. Trümper paper given in the link) 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 start numbers M(n, k) for Collatz sequences realizing the Collatz word (udd)^n ud = (sd)^n s (s = ud is useful because, except for the one letter word u, at least one d follows a letter u), with n >= 1, and k >= 1. The length of these Collatz sequences 3*n. For these Collatz sequences M(n, 0) = M(1, 0) = 1 and N(n, 0) = N(1, 0) = 2.

Examples

			The rectangular array M(n, k) begins:
n\k 0       1       2       3       4       5 ...
1:  1       3       5       7       9      11
2:  1       9      17      25      33      41
3:  1      33      65      97     129     161
4:  1     129     257     385     513     641
5:  1     513    1025    1537    2049    2561
6:  1    2049    4097    6145    8193   10241
7:  1    8193   16385   24577   32769   40961
8:  1   32769   65537   98305  131073  163841
9:  1  131073  262145  393217  524289  655361
10: 1  524289 1048577 1572865 2097153 2621441
...
For more columns see the link.
The triangle TM(m, n) begins (zeros are not shown):
k\n  1  2   3   4    5     6      7 ...
0:   1
1:   3  1
2:   5  9   1
3:   7 17  33   1
4:   9 25  65 129    1
5:  11 33  97 257  513     1
6:  13 41 129 385 1025  2049      1
...
For more rows see the link.
n=1, ud, k=0: M(1, 0) = 1 = TM(0, 1), N(1, 0) = 2 with the Collatz sequence [1, 4, 2] of
length 3.
n=1, ud, k=2: M(1, 2) = 5 = TM(2, 1), N(1,  2) = 8 with the Collatz sequence [5, 16, 8] of length 3.
n=2, uddud, k=0: M(2, 0) = 1 = TM(1, 2), Ne(2, 0) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9.
		

Links

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

Crossrefs

Cf. A238475, A238476, A239126, A239127.

Formula

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

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