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A238476
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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.
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7
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3, 7, 13, 11, 29, 53, 15, 45, 117, 213, 19, 61, 181, 469, 853, 23, 77, 245, 725, 1877, 3413, 27, 93, 309, 981, 2901, 7509, 13653, 31, 109, 373, 1237, 3925, 11605, 30037, 54613, 35, 125, 437, 1493, 4949, 15701, 46421, 120149, 218453
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OFFSET
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1,1
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COMMENTS
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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.
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.
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LINKS
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FORMULA
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Mo(n, k) = 2^(2*n)*k - (2^(2*n-1)+1)/3 for n >= 1 and k >= 1.
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.
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EXAMPLE
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The rectangular array Mo(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 ...
1: 3 7 11 15 19 23 27 31 35 39
2: 13 29 45 61 77 93 109 125 141 157
3: 53 117 181 245 309 373 437 501 565 629
4: 213 469 725 981 1237 1493 1749 2005 2261 2517
5: 853 1877 2901 3925 4949 5973 6997 8021 9045 10069
6: 3413 7509 11605 15701 19797 23893 27989 32085 36181 40277
7: 13653 30037 46421 62805 79189 95573 111957 128341 144725 161109
8: 54613 120149 185685 251221 316757 382293 447829 513365 578901 644437
9: 218453 480597 742741 1004885 1267029 1529173 1791317 2053461 2315605 2577749
10: 873813 1922389 2970965 4019541 5068117 6116693 7165269 8213845 9262421 10310997
...
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The triangle To(m, n) begins (zeros are not shown):
m\n 1 2 3 4 5 6 7 8 9 10 ...
1: 3
2: 7 13
3: 11 29 53
4: 15 45 117 213
5: 19 61 181 469 853
6: 23 77 245 725 1877 3413
7: 27 93 309 981 2901 7509 13653
8: 31 109 373 1237 3925 11605 30037 54613
9: 35 125 437 1493 4949 15701 46421 120149 218453
10: 39 141 501 1749 5973 19797 62805 185685 480597 873813
...
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.
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.
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.
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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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