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%I #39 Jun 30 2021 10:39:29
%S 3,7,7,11,15,15,15,23,31,31,19,31,47,63,63,23,39,63,95,127,127,27,47,
%T 79,127,191,255,255,31,55,95,159,255,383,511,511,35,63,111,191,319,
%U 511,767,1023,1023,39,71,127,223,383,639,1023,1535,2047,2047
%N Rectangular array showing the starting values M(n, k), k >= 1, for the Collatz operation (ud)^n, n >= 1, ending in an odd number, read by antidiagonals.
%C The companion array and triangle for the odd end numbers N(n, k) is given in A239127.
%C 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 the Collatz word (ud)^n = s^n (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. Such Collatz sequences have the maximal number of u's (grow fastest).
%C This rectangular array is M of Example 2.2. with x=y = n, n >= 1, of the M. Trümper reference, pp. 7-8, written as a triangle by taking NE-SW diagonals. The Collatz sequence starting with M(n, k) has length 2*n+1 for each k and it ends in the odd number N(n, k) given in A239127.
%C The first row sequences of the array M (columns of triangle TM) are A004767, A004771, A125169, A239128, ...
%H Wolfdieter 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 The array: M(n, k) = 2^(n+1)*k - 1 for n >= 1 and k >= 1.
%F The triangle: TM(m, n) = M(n, m-n+1) = 2^(n+1)*(m-n+1) - 1 for m >= n >= 1 and 0 for m < n.
%F a(n) = 4*A087808(A130328(n-1)) - 1 (conjectured). - _Christian Krause_, Jun 15 2021
%e The rectangular array M(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: 7 15 23 31 39 47 55 63 71 79
%e 3: 15 31 47 63 79 95 111 127 143 159
%e 4: 31 63 95 127 159 191 223 255 287 319
%e 5: 63 127 191 255 319 383 447 511 575 639
%e 6: 127 255 383 511 639 767 895 1023 1151 1279
%e 7: 255 511 767 1023 1279 1535 1791 2047 2303 2559
%e 8: 511 1023 1535 2047 2559 3071 3583 4095 4607 5119
%e 9: 1023 2047 3071 4095 5119 6143 7167 8191 9215 10239
%e 10: 2047 4095 6143 8191 10239 12287 14335 16383 18431 20479
%e ...
%e The triangle TM(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 7
%e 3: 11 15 15
%e 4: 15 23 31 31
%e 5: 19 31 47 63 63
%e 6: 23 39 63 95 127 127
%e 7: 27 47 79 127 191 255 255
%e 8: 31 55 95 159 255 383 511 511
%e 9: 35 63 111 191 319 511 767 1023 1023
%e 10: 39 71 127 223 383 639 1023 1535 2047 2047
%e ...
%e ---------------------------------------------------------------------
%e n=1, ud, k=1: M(1, 1) = 3 = TM(1, 1), N(1,1) = 5 with the Collatz sequence [3, 10, 5] of length 3.
%e n=1, ud, k=2: M(1, 2) = 7 = TM(2, 1), N(1,2) = 11 with the Collatz sequence [7, 22, 11] of length 3.
%e n=4, (ud)^4, k=2: M(4, 2) = 63 = TM(5, 4), N(4,2) = 323 with the Collatz sequence [63, 190, 95, 286, 143, 430, 215, 646, 323] of length 9.
%e n=5, (ud)^5, k=1: M(5, 1) = 63 = TM(5, 5), N(5,1) = 485 with the Collatz sequence [63, 190, 95, 286, 143, 430, 215, 646, 323, 970, 485] of length 11.
%Y Cf. A006577, A139399, A112695, A238475, A238476, A004767, A004771, A125169, A239127, A239128.
%K nonn,tabl,easy
%O 1,1
%A _Wolfdieter Lang_, Mar 13 2014
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