Self Organizing Sequence

Number Theory Level 4

Consider the sequence with the following properties.

The first term $a_1$ is any natural number.

The subsequent numbers follow the rule

$a_{n+1}=\left\{\begin{array}{lr}a_n/2 & if\ a_n\ is\ even\\ 3a_n+1 & if\ a_n\ is\ odd \end{array}\right.$

What is the value of $\lim_{n \to \infty} \frac{\sum_{i=1}^n a_i}{n}$

The answer is 2.3333.

2 solutions

Janardhanan Sivaramakrishnan
Feb 13, 2015

The sequence ultimately converges to the repeating series ...,4,2,1,4,2,1,4,2,1...

So, at very large $n$ the average value tends to the average of the above three numbers.

Hence, the required answer is $\frac{4+2+1}{3}=\boxed{2.333}$

That never happens. The series would actually look like ...,4,2,7,3.5,10.5,...

Amar Shah - 5 years, 10 months ago

Let us say for a large $n$ , $a_n=4$

Then, as 4 is even $a_{n+1} = 4/2 =2$ and since 2 is 'even' $a_{n+2} = 2/2 =1$ Now, 1 is odd, hence $a_{n+3} = 3*1+1 = 4$

You seem to have swapped the formula for even and odd $a_n$ .

Janardhanan Sivaramakrishnan - 5 years, 10 months ago

Isn't this based off the unproven Collatz conjecture?

Jared Low - 5 years, 8 months ago

Brock Brown
Feb 18, 2015

Python:

from fractions import Fraction as frac
memo = {1:1}
def a(n):
    if n in memo:
        return memo[n]
    if a(n-1) % 2 == 0:
        memo[n] = frac(a(n-1),2)
    else:
        memo[n] = 3*a(n-1)+1
    return memo[n]
def f(n):
    total = 0
    for i in xrange(1, n+1):
        total += a(i)
    return frac(total,n)
infinity = 150
print "Answer:", float(f(infinity))

Self Organizing Sequence

The answer is 2.3333.

2 solutions

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