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.
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
Examples
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.
Links
- Wolfdieter Lang, Rectangular array and triangle.
- 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.
- Wikipedia, Collatz Conjecture
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.
Comments