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%I #32 Jul 25 2020 21:17:58
%S 3,7,13,11,29,53,15,45,117,213,19,61,181,469,853,23,77,245,725,1877,
%T 3413,27,93,309,981,2901,7509,13653,31,109,373,1237,3925,11605,30037,
%U 54613,35,125,437,1493,4949,15701,46421,120149,218453
%N 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.
%C 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.
%C 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.
%C The first rows of array Mo (columns of triangle To) are A004767, A082285, A239124, ...
%H W. Lang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Lang/lang6.html">On Collatz' Words, Sequences, and Trees</a>, J. of Integer Sequences, Vol. 17 (2014), Article 14.11.7.
%H Manfred Trümper, <a href="http://dx.doi.org/10.1155/2014/756917">The Collatz Problem in the Light of an Infinite Free Semigroup</a>, Chinese Journal of Mathematics, Vol. 2014, Article ID 756917, 21 pages.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Collatz_conjecture">Collatz Conjecture</a>.
%F Mo(n, k) = 2^(2*n)*k - (2^(2*n-1)+1)/3 for n >= 1 and k >= 1.
%F 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.
%e The rectangular array Mo(n, k) begins:
%e n\k 1 2 3 4 5 6 7 8 9 10 ...
%e 1: 3 7 11 15 19 23 27 31 35 39
%e 2: 13 29 45 61 77 93 109 125 141 157
%e 3: 53 117 181 245 309 373 437 501 565 629
%e 4: 213 469 725 981 1237 1493 1749 2005 2261 2517
%e 5: 853 1877 2901 3925 4949 5973 6997 8021 9045 10069
%e 6: 3413 7509 11605 15701 19797 23893 27989 32085 36181 40277
%e 7: 13653 30037 46421 62805 79189 95573 111957 128341 144725 161109
%e 8: 54613 120149 185685 251221 316757 382293 447829 513365 578901 644437
%e 9: 218453 480597 742741 1004885 1267029 1529173 1791317 2053461 2315605 2577749
%e 10: 873813 1922389 2970965 4019541 5068117 6116693 7165269 8213845 9262421 10310997
%e ...
%e ---------------------------------------------------------------------------------------------
%e The triangle To(m, n) begins (zeros are not shown):
%e m\n 1 2 3 4 5 6 7 8 9 10 ...
%e 1: 3
%e 2: 7 13
%e 3: 11 29 53
%e 4: 15 45 117 213
%e 5: 19 61 181 469 853
%e 6: 23 77 245 725 1877 3413
%e 7: 27 93 309 981 2901 7509 13653
%e 8: 31 109 373 1237 3925 11605 30037 54613
%e 9: 35 125 437 1493 4949 15701 46421 120149 218453
%e 10: 39 141 501 1749 5973 19797 62805 185685 480597 873813
%e ...
%e 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.
%e 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.
%e 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.
%Y Cf. A006577, A139399, A112695, A238475, A004767, A082285, A239124.
%K nonn,tabl,easy
%O 1,1
%A _Wolfdieter Lang_, Mar 10 2014
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