OFFSET
1
COMMENTS
The 5x+1 function (A185452), denoted by T_5(x) in the literature, is defined as T_5(x) = (5x+1)/2 if x is odd, T_5(x) = x/2 if x is even.
As reported by Kontorovich and Lagarias (2009 and 2010), and analogously to A365495, the sequence of the first m terms in each row is periodic in n with period 2^m, with each of the 2^m possible binary vectors occurring exactly once (as the first m terms of a row) per period.
For example, for m = 3, the first 3 terms in rows 1..2^3 are respectively [1,1,0], [0,1,1], [1,0,0], [0,0,1], [1,1,1], [0,1,0], [1,0,1] and [0,0,0], and this pattern repeats from row 2^3 + 1 onwards.
As a consequence, Kontorovich and Lagarias note, each integer is uniquely determined by the sequence of the parity of its orbit, i.e., n is uniquely determined by the n-th row of the present array.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150 of the array, flattened)
Alex V. Kontorovich and Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems, arXiv:0910.1944 [math.NT], 2009, pp. 39-40, and in Jeffrey C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, pp. 164-165.
FORMULA
T(n,k) = A365991(n,k) mod 2.
EXAMPLE
The array begins:
n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
-----------------------------------------------------
1 | 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, ...
2 | 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, ...
3 | 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, ...
4 | 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, ...
5 | 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, ...
6 | 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, ...
7 | 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, ...
8 | 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, ...
9 | 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, ...
10 | 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, ...
11 | 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, ...
12 | 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, ...
13 | 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, ...
14 | 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, ...
15 | 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, ...
...
MATHEMATICA
A365992list[dmax_]:=With[{a=Mod[Array[NestList[If[OddQ[#], (5#+1)/2, #/2]&, dmax-#, #]&, dmax, 0], 2]}, Array[Diagonal[a, #]&, dmax, 1-dmax]]; A365992list[20] (* Generates 20 antidiagonals *)
CROSSREFS
KEYWORD
AUTHOR
Paolo Xausa, Sep 25 2023
STATUS
approved