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%I #22 Sep 01 2018 21:44:27
%S 5,11,17,17,35,53,23,53,107,161,29,71,161,323,485,35,89,215,485,971,
%T 1457,41,107,269,647,1457,2915,4373,47,125,323,809,1943,4373,8747,
%U 13121,53,143,377,971,2429,5831,13121,26243,39365,59,161,431,1133,2915,7289,17495,39365,78731,118097
%N Rectangular companion array to M(n,k), given in A239126, showing the end numbers N(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 start numbers M(n, k) is given in A239126.
%C See the comments on A239126 for the Collatz 3x+1 problem and the u and d operations.
%C This rectangular array is N of the 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 odd M(n, k) from A239126 and ending in odd N(n, k) has length 2*n+1 for each k.
%C The first row sequences of the array N (columns of triangle TN) are A016969, A239129, ...
%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.
%F The array: N(n, k) = 2*3^n*k - 1 for n >= 1 and k >= 1.
%F The triangle: TN(m, n) = N(n, m-n+1) = 2*3^n*(m-n+1) - 1 for m >= n >= 1 and 0 for m < n.
%e The rectangular array N(n, k) begins:
%e n\k 1 2 3 4 5 6 7 8 9 10 ...
%e 1: 5 11 17 23 29 35 41 47 53 59
%e 2: 17 35 53 71 89 107 125 143 161 179
%e 3: 53 107 161 215 269 323 377 431 485 539
%e 4: 161 323 485 647 809 971 1133 1295 1457 1619
%e 5: 485 971 1457 1943 2429 2915 3401 3887 4373 4859
%e 6: 1457 2915 4373 5831 7289 8747 10205 11663 13121 14579
%e 7: 4373 8747 13121 17495 21869 26243 30617 34991 39365 43739
%e 8: 13121 26243 39365 52487 65609 78731 91853 104975 118097 131219
%e 9: 39365 78731 118097 157463 196829 236195 275561 314927 354293 393659
%e 10: 118097 236195 354293 472391 590489 708587 826685 944783 1062881 1180979
%e ...
%e -------------------------------------------------------------------------------
%e The triangle TN(m, n) begins (zeros are not shown):
%e m\n 1 2 3 4 5 6 7 8 9 10 ...
%e 1: 5
%e 2: 11 17
%e 3: 17 35 53
%e 4: 23 53 107 161
%e 5: 29 71 161 323 485
%e 6: 35 89 215 485 971 1457
%e 7: 41 107 269 647 1457 2915 4373
%e 8: 47 125 323 809 1943 4373 8747 13121
%e 9: 53 143 377 971 2429 5831 13121 26243 39365
%e 10: 59 161 431 1133 2915 7289 17495 39365 78731 118097
%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, A239126, A016969, A239129.
%K nonn,tabl,easy
%O 1,1
%A _Wolfdieter Lang_, Mar 13 2014
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