OFFSET
0,1
COMMENTS
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.
FORMULA
The array: N(n, k) = 2 + 3^n*k for n >= 1 and k >= 0.
The triangle: TN(m, n) = N(n,m-n+1) = 2 + 3^n*(m-n+1) for m+1 >= n >= 1 and 0 for m+1 < n.
EXAMPLE
The rectangular array N(n, k) begins
n\k 0 1 2 3 4 5 ...
1: 2 5 8 11 14 17
2: 2 11 20 29 38 47
3: 2 29 56 83 110 137
4: 2 83 164 245 326 407
5: 2 245 488 731 974 1217
6: 2 731 1460 2189 2918 3647
7: 2 2189 4376 6563 8750 10937
8: 2 6563 13124 19685 26246 32807
9: 2 19685 39368 59051 78734 98417
10: 2 59051 118100 177149 236198 295247
...
For more columns see the link.
The triangle TN(m, n) begins (zeros are not shown):
m\n 1 2 3 4 5 6 7 ...
0: 2
1: 5 2
2: 8 11 2
3: 11 20 29 2
4: 14 29 56 83 2
5: 17 38 83 164 245 2
6: 20 47 110 245 488 731 2
...
For more rows see the link.
n=1, ud, k=0: M(1, 0) = 1, N(1, 0) = TN(0, 1) = 2 with the Collatz sequence [1, 4, 2] of length 3.
n=1, ud, k=2: M(1, 2) = 5, N(1, 2) = TN(2, 1) = 8 with the Collatz sequence [5, 16, 8] of length 3.
n=2, uddud, k=0: M(2, 0) = 1, Ne(2, 0) = TN(1, 2) = 2 with the Collatz sequence [1, 4, 2, 1, 4, 2, 1, 4, 2] of length 9.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Apr 04 2014
STATUS
approved