OFFSET
1,1
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.
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.
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.
EXAMPLE
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.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Mar 10 2014
STATUS
approved