A329263 Irregular triangle read by rows in which row n is the result of iterating the operation f(n) = n/8 if n == 0 (mod 8), otherwise f(n) = 8*(floor(n/8) + n + 1), terminating at the first occurrence of 1.

1, 2, 24, 3, 32, 4, 40, 5, 48, 6, 56, 7, 64, 8, 1, 3, 32, 4, 40, 5, 48, 6, 56, 7, 64, 8, 1, 4, 40, 5, 48, 6, 56, 7, 64, 8, 1, 5, 48, 6, 56, 7, 64, 8, 1, 6, 56, 7, 64, 8, 1, 7, 64, 8, 1, 8, 1, 9, 88, 11, 104, 13, 120, 15, 136, 17, 160, 20, 184, 23, 208

Offset: 1

Author

Davis Smith, Nov 09 2019

Keywords

• nonn
• easy
• tabf

Comments

The operation f(n) can be generalized to C(n,m) = n/m if n == 0 (mod m), m*(floor(n/m) + n + 1) otherwise. The operation for the 3x+1 (Collatz) problem is equivalent to C(n,2) and f(n) = C(n,8).

Conjecture: For any initial value of n >= 1 there is a number k such that f^{k}(n) = 1; in other words, every row of the triangle is finite.

Examples

			The irregular array a(n,k) starts:
n\k   0   1   2   3   4    5   6    7   8    9  10  11  12  13 ...
1:    1
2:    2  24   3  32   4   40   5   48   6   56   7  64   8   1
3:    3  32   4  40   5   48   6   56   7   64   8   1
4:    4  40   5  48   6   56   7   64   8    1
5:    5  48   6  56   7   64   8    1
6:    6  56   7  64   8    1
7:    7  64   8   1
8:    8   1
9:    9  88  11 104  13  120  15  136  17  160  20 184  23 208 ...
10:  10  96  12 112  14  128  16    2  24    3  32   4  40   5 ...
a(9,100) = 1 and a(10,20) = 1.
		

Crossrefs

Cf. A070165.

Programs

PARI

collatz8(n)=N=[n];while(n>1,N=concat(N,n=if(n%8,8*(floor(n/8)+n+1),n/8)));N

Formula

a(n,0) = n, a(n,k + 1) = a(n,k)/8 if a(n,k) == 0 (mod 8), 8*(floor(a(n,k)/8) + a(n,k) + 1) otherwise, for n >= 1.