A001281 Image of n under the map n->n/2 if n even, n->3n-1 if n odd.

0, 2, 1, 8, 2, 14, 3, 20, 4, 26, 5, 32, 6, 38, 7, 44, 8, 50, 9, 56, 10, 62, 11, 68, 12, 74, 13, 80, 14, 86, 15, 92, 16, 98, 17, 104, 18, 110, 19, 116, 20, 122, 21, 128, 22, 134, 23, 140, 24, 146, 25, 152, 26, 158, 27, 164, 28, 170, 29, 176, 30, 182, 31, 188

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

On the set of positive integers, the orbit of any number seems to end in the orbit of 1, of 5 or of 17. Writing n=1+q*2^p with q odd, it is easily seen that for p=0,1 and p>3, some iterations of the map lead to a strictly smaller number (for n>17). The cases p=2 and p=3 may give rise to bigger loops (depending on the form of q). See sequences A135727-A135729 for maxima of the orbits and corresponding record indices. - M. F. Hasler, Nov 29 2007

References

• R. K. Guy, Unsolved Problems in Number Theory, E16.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Crossrefs

Cf. A037082.

Cf. A037084, A039500-A039505, A135727-A135730. See also A006370, A006577 (Collatz 3x+1 problem).

Programs

Maple

f := n-> if n mod 2 = 0 then n/2 else 3*n-1; fi;

Mathematica

Table[If[OddQ[n], 3*n-1, n/2], {n, 0, 100}] (* T. D. Noe, Jun 27 2012 *)

PARI

A001281(n)=if(n%2,3*n-1,n>>1) \\ M. F. Hasler, Nov 29 2007

Formula

f(n) = (7n-2-(5n-2)*cos(Pi*n))/4. - Robert W. Craigen (craigen(AT)fresno.edu)

G.f.: x*(2 + x + 4*x^2)/((1 - x)^2*(1 + x)^2). - Ilya Gutkovskiy, Aug 17 2016