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A240222
Rectangular array giving all start values M(n, k), k >= 1, for Collatz sequences following the pattern (udd)^(n-1) ud, n >= 1, read by antidiagonals.
2
1, 3, 1, 5, 9, 1, 7, 17, 33, 1, 9, 25, 65, 129, 1, 11, 33, 97, 257, 513, 1, 13, 41, 129, 385, 1025, 2049, 1, 15, 49, 161, 513, 1537, 4097, 8193, 1, 17, 57, 193, 641, 2049, 6145, 16385, 32769, 1, 19, 65, 225, 769, 2561, 8193, 24577, 65537, 131073, 1, 21, 73, 257, 897, 3073, 10241, 32769, 98305
OFFSET
1,2
COMMENTS
The companion array and triangle for the end numbers N(n, k) is given in A240223.
The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here (following the M. Trümper paper given in the link) 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 start numbers M(n, k) for Collatz sequences realizing the Collatz word (udd)^n ud = (sd)^n s (s = ud is useful because, except for the one letter word u, at least one d follows a letter u), with n >= 1, and k >= 1. The length of these Collatz sequences 3*n. For these Collatz sequences M(n, 0) = M(1, 0) = 1 and N(n, 0) = N(1, 0) = 2.
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.
Eric Weisstein's World of Mathematics, Collatz Problem.
FORMULA
The array: M(n, k) = 1 + 2^(2*n-1)*k for n >= 1 and k >= 0.
The triangle: TM(m, n) = M(n,m-n+1) = 1 + 2^(2*n-1)*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n.
EXAMPLE
The rectangular array M(n, k) begins:
n\k 0 1 2 3 4 5 ...
1: 1 3 5 7 9 11
2: 1 9 17 25 33 41
3: 1 33 65 97 129 161
4: 1 129 257 385 513 641
5: 1 513 1025 1537 2049 2561
6: 1 2049 4097 6145 8193 10241
7: 1 8193 16385 24577 32769 40961
8: 1 32769 65537 98305 131073 163841
9: 1 131073 262145 393217 524289 655361
10: 1 524289 1048577 1572865 2097153 2621441
...
For more columns see the link.
The triangle TM(m, n) begins (zeros are not shown):
k\n 1 2 3 4 5 6 7 ...
0: 1
1: 3 1
2: 5 9 1
3: 7 17 33 1
4: 9 25 65 129 1
5: 11 33 97 257 513 1
6: 13 41 129 385 1025 2049 1
...
For more rows see the link.
n=1, ud, k=0: M(1, 0) = 1 = TM(0, 1), N(1, 0) = 2 with the Collatz sequence [1, 4, 2] of
length 3.
n=1, ud, k=2: M(1, 2) = 5 = TM(2, 1), N(1, 2) = 8 with the Collatz sequence [5, 16, 8] of length 3.
n=2, uddud, k=0: M(2, 0) = 1 = TM(1, 2), Ne(2, 0) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9.
CROSSREFS
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Apr 02 2014
STATUS
approved