Puzzling Asked on May 20, 2021
The sequence starts with the following:
a1 = 1, a2 = 2, a3 = 5, a4 = 4, a5 = 6,
a6 = 10, a7 = 9, a8 = 8, a9 = 21, a10 = 12,
a11 = 13, a12 = 20, a13 = 33, a14 = 15, a15 = 42,
a16 = 16, a17 = 19, a18 = 63, a19 = 34, a20 = 24, …
Identify the rule of this sequence and the next five terms.
The same list of numbers, without indices:
1, 2, 5, 4, 6, 10, 9, 8, 21, 12, 13, 20, 33, 15, 42, 16, 19, 63, 34, 24
Hints are spoilered so that people can choose to solve it without seeing them.
Hint 1
Hint 2
Hint 3-1 (continuation of Hint 1)
Hint 3-2 (continuation of Hint 2)
It is very clear that -
Now,
Further,
Next,
Other sequences
From here I got confused and couldn't get it
Answered by AnilGoyal on May 20, 2021
Before I start, consider function $f$ that satisfies $A_x = f(x)$. That is, to find the $x$-th number in the sequence, we can plug in $x$ to $f$.
So we want to find the values of $f(21), f(22), f(23), f(24), f(25)$.
Inspired from hint 2, instead of mapping x to $f(x)$, what if we try to map the relation of $x$ and $f(f(x))$ instead? We get this following pattern.
$x$ | $f(x)$ | $f(f(x))$ | $x$ in binary |
---|---|---|---|
$1$ | $f(1)=1$ | $f(1)=1=1times1$ | $1$ |
$2$ | $f(2)=2$ | $f(2)=2=1times2$ | $10$ |
$3$ | $f(3)=5$ | $f(5)=6=2times3$ | $11$ |
$4$ | $f(4)=4$ | $f(4)=4=1times4$ | $100$ |
$5$ | $f(5)=6$ | $f(6)=10=2times5$ | $101$ |
$6$ | $10$ | $12=2times6$ | $110$ |
$7$ | $9$ | $21=3times7$ | $111$ |
$8$ | $8$ | $8=1times8$ | $1000$ |
$9$ | $21$ | $18=2times9$ | $1001$ |
and so on. Based on the provided sequence, it can be said that
That is possibly why $f(f(5))=10=2times5$ but $f(f(7))=21=3times7$.
BONUS:
From rearranging the function we can obtain a new relation
$f(x)=f^{-1}(n_xtimes x)$.
For the values of $x=21,22,23,24,25$, it can be calculated that $n_{21}=3, n_{22}=3, n_{23}=4,n_{24}=2,n_{25}=3$.
Therefore,
Finding $f(24)$ is fun because if we try to construct $x, f(x), f(f(x)), f(f(f(x))), ...$ starting from $x=3$, we get the sequence
So far I can't use $f(22)=f^{-1}(66)$ and $f(25)=f^{-1}(75)$ to my advantage and I'm stuck..
However, I made the connection that the 'root' in title and hint 1 is perhaps meaning 'half-iterate' or 'functional square root', see this Wikipedia page.
Answered by Nuclear03020704 on May 20, 2021
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