A238475 - OEIS

%I #31 Jul 28 2020 17:51:22

%S 1,9,5,17,37,21,25,69,149,85,33,101,277,597,341,41,133,405,1109,2389,

%T 1365,49,165,533,1621,4437,9557,5461,57,197,661,2133,6485,17749,38229,

%U 21845,65,229,789,2645,8533,25941,70997,152917,87381

%N 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.

%C 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.

%C 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.

%C The first row sequences of the array Me (they become columns in the triangle Te) are A017077, A238477, A239123, ...

%C 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.

%H W. 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Collatz_conjecture">Collatz Conjecture</a>.

%F The array: Me(n, k) = 2^(2*n+1)*k - (5*2^(2*n)+1)/3 for n >= 1 and k >= 1.

%F 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.

%e The rectangular array Me(n, k) begins:

%e n\k      1       2       3        4       5        6        7        8        9       10 ...

%e 1:       1       9      17       25      33       41       49       57       65       73

%e 2:       5      37      69      101     133      165      197      229      261      293

%e 3:      21     149     277      405     533      661      789      917     1045     1173

%e 4:      85     597    1109     1621    2133     2645     3157     3669     4181     4693

%e 5:     341    2389    4437     6485    8533    10581    12629    14677    16725    18773

%e 6:    1365    9557   17749    25941   34133    42325    50517    58709    66901    75093

%e 7:    5461   38229   70997   103765  136533   169301   202069   234837   267605   300373

%e 8:   21845  152917  283989   415061  546133   677205   808277   939349  1070421  1201493

%e 9:   87381  611669 1135957  1660245 2184533  2708821  3233109  3757397  4281685  4805973

%e 10: 349525 2446677 4543829  6640981 8738133 10835285 12932437 15029589 17126741 19223893

%e ...

%e The triangle Te(m, n) begins (zeros are not shown):

%e m\n   1    2    3     4      5      6       7       8       9      10 ...

%e 1:    1

%e 2:    9    5

%e 3:   17   37   21

%e 4:   25   69  149    85

%e 5:   33  101  277   597    341

%e 6:   41  133  405  1109   2389   1365

%e 7:   49  165  533  1621   4437   9557    5461

%e 8:   57  197  661  2133   6485  17749   38229   21845

%e 9:   65  229  789  2645   8533  25941   70997  152917   87381

%e 10:  73  261  917  3157  10581  34133  103765  283989  611669  349525

%e ...

%e ----------------------------------------------------------------------------------------------

%e 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.

%e 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.

%e 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].

%e 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.

%Y Cf. A006577, A139399, A112695, A238476, A017077, A238477, A239123.

%K nonn,easy,tabl

%O 1,2

%A _Wolfdieter Lang_, Mar 10 2014