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A239127
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.
6
5, 11, 17, 17, 35, 53, 23, 53, 107, 161, 29, 71, 161, 323, 485, 35, 89, 215, 485, 971, 1457, 41, 107, 269, 647, 1457, 2915, 4373, 47, 125, 323, 809, 1943, 4373, 8747, 13121, 53, 143, 377, 971, 2429, 5831, 13121, 26243, 39365, 59, 161, 431, 1133, 2915, 7289, 17495, 39365, 78731, 118097
OFFSET
1,1
COMMENTS
The companion array and triangle for the odd start numbers M(n, k) is given in A239126.
See the comments on A239126 for the Collatz 3x+1 problem and the u and d operations.
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.
The first row sequences of the array N (columns of triangle TN) are A016969, A239129, ...
LINKS
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.
FORMULA
The array: N(n, k) = 2*3^n*k - 1 for n >= 1 and k >= 1.
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.
EXAMPLE
The rectangular array N(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 ...
1: 5 11 17 23 29 35 41 47 53 59
2: 17 35 53 71 89 107 125 143 161 179
3: 53 107 161 215 269 323 377 431 485 539
4: 161 323 485 647 809 971 1133 1295 1457 1619
5: 485 971 1457 1943 2429 2915 3401 3887 4373 4859
6: 1457 2915 4373 5831 7289 8747 10205 11663 13121 14579
7: 4373 8747 13121 17495 21869 26243 30617 34991 39365 43739
8: 13121 26243 39365 52487 65609 78731 91853 104975 118097 131219
9: 39365 78731 118097 157463 196829 236195 275561 314927 354293 393659
10: 118097 236195 354293 472391 590489 708587 826685 944783 1062881 1180979
...
-------------------------------------------------------------------------------
The triangle TN(m, n) begins (zeros are not shown):
m\n 1 2 3 4 5 6 7 8 9 10 ...
1: 5
2: 11 17
3: 17 35 53
4: 23 53 107 161
5: 29 71 161 323 485
6: 35 89 215 485 971 1457
7: 41 107 269 647 1457 2915 4373
8: 47 125 323 809 1943 4373 8747 13121
9: 53 143 377 971 2429 5831 13121 26243 39365
10: 59 161 431 1133 2915 7289 17495 39365 78731 118097
...
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.
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.
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.
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.
CROSSREFS
KEYWORD
nonn,tabl,easy
AUTHOR
Wolfdieter Lang, Mar 13 2014
STATUS
approved