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A238475
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Rectangular array with all start numbers Me(n, k), k >= 1, for the Collatz operation ud^(2*n), n >= 1, ending in an odd number, read by antidiagonals.
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9
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1, 9, 5, 17, 37, 21, 25, 69, 149, 85, 33, 101, 277, 597, 341, 41, 133, 405, 1109, 2389, 1365, 49, 165, 533, 1621, 4437, 9557, 5461, 57, 197, 661, 2133, 6485, 17749, 38229, 21845, 65, 229, 789, 2645, 8533, 25941, 70997, 152917, 87381
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OFFSET
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1,2
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COMMENTS
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The two operations on natural numbers m used in the Collatz 3x+1 conjecture (see the links) are here (following the M. Trümper reference) 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 positive start numbers Me(n, k), k >= 1, for Collatz sequences following the pattern (word) ud^(2*n), for n >= 1, which end in an odd number. The end number does not depend on n and it is given by Ne(k) = 6*k - 5.
This rectangular array is Example 2.1. with x = 2*n, n >= 1, of the M. Trümper reference, pp. 4-5, written as a triangle by taking NE-SW diagonals. The case x = 2*n+1, n >= 0, for the word ud^(2*k+1) appears as array and triangle in A238476.
The first row sequences of the array Me (they become columns in the triangle Te) are A017077, A238477, A239123, ...
Note that there are also Collatz sequences starting with an odd number, following the pattern ud^(2*n) which end in an even number. For example, take n=1 and the sequence [5, 16, 8, 4]. Such sequences are here not considered.
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LINKS
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FORMULA
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The array: Me(n, k) = 2^(2*n+1)*k - (5*2^(2*n)+1)/3 for n >= 1 and k >= 1.
The triangle: Te(m, n) = Me(n, m-n+1) = 2*4^n*(m-n) + (4^n-1)/3 for m >= n >= 1 and 0 for m < n.
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EXAMPLE
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The rectangular array Me(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 ...
1: 1 9 17 25 33 41 49 57 65 73
2: 5 37 69 101 133 165 197 229 261 293
3: 21 149 277 405 533 661 789 917 1045 1173
4: 85 597 1109 1621 2133 2645 3157 3669 4181 4693
5: 341 2389 4437 6485 8533 10581 12629 14677 16725 18773
6: 1365 9557 17749 25941 34133 42325 50517 58709 66901 75093
7: 5461 38229 70997 103765 136533 169301 202069 234837 267605 300373
8: 21845 152917 283989 415061 546133 677205 808277 939349 1070421 1201493
9: 87381 611669 1135957 1660245 2184533 2708821 3233109 3757397 4281685 4805973
10: 349525 2446677 4543829 6640981 8738133 10835285 12932437 15029589 17126741 19223893
...
The triangle Te(m, n) begins (zeros are not shown):
m\n 1 2 3 4 5 6 7 8 9 10 ...
1: 1
2: 9 5
3: 17 37 21
4: 25 69 149 85
5: 33 101 277 597 341
6: 41 133 405 1109 2389 1365
7: 49 165 533 1621 4437 9557 5461
8: 57 197 661 2133 6485 17749 38229 21845
9: 65 229 789 2645 8533 25941 70997 152917 87381
10: 73 261 917 3157 10581 34133 103765 283989 611669 349525
...
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n=1, ud^2, k=1: Me(1, 1) = 1 = Te(1, 1), Ne(1) = 1 with the Collatz sequence [1, 4, 2, 1] of length 4.
n=1, ud^2, k=2: Me(1, 2) = 9 = Te(2, 1), Ne(2) = 7 with the Collatz sequence [9, 28, 14, 7] of length 4.
n=2, ud^4, k=1: Me(2, 1) = 5 = Te(2, 2), Ne(1) = 1 with the length 6 Collatz sequence [5, 16, 8, 4, 2, 1].
n=5, ud^(10), k=2: Me(5, 2) = 2389 = Te(6,5), Ne(2) = 7 with the Collatz sequence [2389, 7168, 3584, 1792, 896, 448, 224, 112, 56, 28, 14, 7] of length 12.
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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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