A288251 k-th term of n in the 3x+1 problem, k >= 1. Square array A(n, k) read by antidiagonals downwards.

4, 2, 1, 1, 4, 10, 4, 2, 5, 2, 2, 1, 16, 1, 16, 1, 4, 8, 4, 8, 3, 4, 2, 4, 2, 4, 10, 22, 2, 1, 2, 1, 2, 5, 11, 4, 1, 4, 1, 4, 1, 16, 34, 2, 28, 4, 2, 4, 2, 4, 8, 17, 1, 14, 5, 2, 1, 2, 1, 2, 4, 52, 4, 7, 16, 34, 1, 4, 1, 4, 1, 2, 26, 2, 22, 8, 17, 6, 4, 2, 4, 2, 4, 1, 13, 1, 11, 4, 52, 3, 40

Offset

1,1

Comments

This is the A347270 array without the first column. - Paolo Xausa, Jan 18 2026

Links

Alois P. Heinz, Antidiagonals n = 1..200, flattened

Wikipedia, Collatz conjecture

Index entries for sequences related to 3x+1 problem

Formula

T(m, n) = A(n, m-n+1), 1 <= n <= m, with A(n, k) = C^{[k]}(n), n >= 1, k >= 1, with iterations of the Collatz map C: n -> 3*n+1 if n is odd and n -> n/2 if n is even. - Wolfdieter Lang, Jul 13 2017

Example

The array A(n, k) starts:
  n\k |  1  2  3  4  5  6  7  8  9 10 11 12 ...
  ---------------------------------------------
  1   |  4  2  1  4  2  1  4  2  1  4  2  1
  2   |  1  4  2  1  4  2  1  4  2  1  4  2
  3   | 10  5 16  8  4  2  1  4  2  1  4  2
  4   |  2  1  4  2  1  4  2  1  4  2  1  4
  5   | 16  8  4  2  1  4  2  1  4  2  1  4
  6   |  3 10  5 16  8  4  2  1  4  2  1  4
  7   | 22 11 34 17 52 26 13 40 20 10  5 16
  8   |  4  2  1  4  2  1  4  2  1  4  2  1
  9   | 28 14  7 22 11 34 17 52 26 13 40 20
  10  |  5 16  8  4  2  1  4  2  1  4  2  1
  ...
The triangle T(m, n) starts:
  m\n |  1 2  3 4  5  6  7 8  9 10 ...
  ------------------------------------
  1   |  4
  2   |  2 1
  3   |  1 4 10
  4   |  4 2  5 2
  5   |  2 1 16 1 16
  6   |  1 4  8 4  8  3
  7   |  4 2  4 2  4 10 22
  8   |  2 1  2 1  2  5 11 4
  9   |  1 4  1 4  1 16 34 2 28
  10  |  4 2  4 2  4  8 17 1 14  5
  ... formatted, Wolfdieter Lang, Jul 13 2017

Maple

A:= proc(n, k) option remember; `if`(k=0, n,
      A(`if`(n::even, n/2, 3*n+1), k-1))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Jan 17 2026

Mathematica

A288251list[dmax_] := With[{a = Array[Rest[NestList[If[OddQ[#], 3*# + 1, #/2] &, dmax - # + 1, #]] &, dmax]}, Array[Reverse[Diagonal[a, #]] &, dmax, 1 - dmax]];
A288251list[15] (* Paolo Xausa, Jan 18 2026 *)

Prog

(PARI) trajectory(n, terms) = my(i=0, x=n); while(1, if(x%2==0, x=x/2, x=3*x+1); print1(x, ", "); i++; if(i==terms, break))
array(n, k) = for(x=1, n, trajectory(x, k); print(""))
array(10, 12) \\ print initial 10 rows and 12 columns of array

Crossrefs

Rows of A (columns of T): A153727 (row 1), A033478 (row 3), A033479 (row 9).

Columns of A (diagonals of T): A006370 (column 1), A075884 (column 2), A076536 (column 3).

Cf. A347270.

Sequence in context: A051149 A293424 A152145 * A024553 A051758 A051568

Adjacent sequences: A288248 A288249 A288250 * A288252 A288253 A288254

Keyword

nonn,easy,look,tabl,changed

Author

Felix Fröhlich, Jun 10 2017

Status

approved