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A239127
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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.
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6
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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
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OFFSET
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1,1
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COMMENTS
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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, ...
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
...
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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