A213185 For the Collatz (3x+1) iteration starting at n, number of integers k such that T(n,k) = T(n-1,k) +1 where T(n,k) is the k-th number of the trajectory (n is not counted).

0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 3, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 3, 4, 0, 0, 2, 1, 2, 0, 1, 2, 0, 0, 1, 1, 1, 0, 2, 3, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 2, 1, 1, 0, 4, 5, 0, 0, 3, 1, 1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 2, 3, 0, 0, 5, 1, 0, 0, 1, 2

Offset

1,7

Comments

T(n,k) is periodic, for example

T(6,3) = 10 and T(7,3) = 11;

T(10,3) = 16 and T(11,3) = 17,

…………..

T(6+4m, 3) = 10 + 6m and T(6+4m+1, 3) = 10 + 6m+1.

.................

Links

Michel Lagneau, Table of n, a(n) for n = 1..10000

Example

a(7) = 2 because the Collatz iterations starting at 7 and 6 are :
T(7,k) =[7, 22, 11, 34, 17, 52,  26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]
T(6,k) =[6, 3, 10, 5, 16, 8,  4, 2, 1] ; the number 7 is not counted
=> T(7,3) = T(6,3) + 1 and T(7,5) = T(6,5) + 1, hence a(7) = 2.

Maple

nn:=200:T:=array(1..nn):U:=array(1..nn): aa:=1:
for i from 1 to nn do:
   U[i]:=0:
od:
      for n from 1 to nn do:
          a:=0:k:=0:x:=n:
               for i from 1 to 10^8 while (x>1) do:
                  if irem(x, 2)=0 then
                  x := x/2:a:=a+1:k:=k+1:T[k]:=x:
              else
                 x := 3*x+1: a := a+1: k:=k+1:T[k]:=x:
               fi:
            od:
           it:=0:
                for j from 1 to min(a, aa) do:
                   if T[j]=U[j]+1 then
                   it:=it+1:
                 else
                 fi:
                od:
                     for m from 1 to a do:
                        U[m]:=T[m]:
                     od:
          aa:=a: printf(`%d, `, it): od:

Crossrefs

Cf. A006577, A213186.

Sequence in context: A025442 A260118 A128582 * A285716 A101606 A257469

Adjacent sequences: A213182 A213183 A213184 * A213186 A213187 A213188

Keyword

nonn

Author

Michel Lagneau, Feb 28 2013

Status

approved