Recursive sequences: $S_0 = 1, S_n = (S_{n−1})^{2} + (S_{n−2})^{2} + · · · + (S_0)^{2}$ for all...
$begingroup$
The question is-
Write down the first five values of each of the following recursive sequences.
$S_0 = 1, S_n = (S_{n−1})^{2} + (S_{n−2})^{2} + · · · + (S_0)^{2}$ for all integers $n ⩾ 1$
I'm not sure if this is correct but this is what I did:
$S_1 = (1)^{2} = 1$
$S_2 = (1)^2 + (1)^2 = 2$
$S_3 = (2)^2 + (1)^2 + (1)^2 = 6$
$S_4 = (6)^2 + (2)^2 + (1)^2 + (1)^2 = 42$
$S_5 = (42)^2 + (6)^2 + (2)^2 + (1)^2 + (1)^2 = 1806$
proof-verification recursion
edited Jan 6 at 6:29
Shubham Johri
5,372718
asked Jan 6 at 5:42
llamaro25llamaro25
528
$endgroup$
|
show 2 more comments
$begingroup$
The question is-
Write down the first five values of each of the following recursive sequences.
$S_0 = 1, S_n = (S_{n−1})^{2} + (S_{n−2})^{2} + · · · + (S_0)^{2}$ for all integers $n ⩾ 1$
I'm not sure if this is correct but this is what I did:
$S_1 = (1)^{2} = 1$
$S_2 = (1)^2 + (1)^2 = 2$
$S_3 = (2)^2 + (1)^2 + (1)^2 = 6$
$S_4 = (6)^2 + (2)^2 + (1)^2 + (1)^2 = 42$
$S_5 = (42)^2 + (6)^2 + (2)^2 + (1)^2 + (1)^2 = 1806$
proof-verification recursion
edited Jan 6 at 6:29
Shubham Johri
5,372718
asked Jan 6 at 5:42
llamaro25llamaro25
528
$endgroup$
4
$begingroup$
Looks fine. What is the question, though? By the way, you can simplify to $S_n=S_{n-1}^2+S_{n-1}$.
$endgroup$
– SmileyCraft
Jan 6 at 5:44
$begingroup$
Thanks. This was the question: Write down the first five values of each of the following recursive sequences. S0 = 1, Sn = (Sn−1)^2 + (Sn−2)^2 + · · · + (S0)^2 for all integers n ⩾ 1
$endgroup$
– llamaro25
Jan 6 at 5:46
1
$begingroup$
@meromero25: Then you answered your own question? Any why is this interesting?
$endgroup$
– David G. Stork
Jan 6 at 5:50
$begingroup$
@DavidG.Stork I wasn't sure if the answer is correct
$endgroup$
– llamaro25
Jan 6 at 5:52
$begingroup$
Just for your curiosity : have a look at oeis.org/… What it seems is that $log(log(S_n))$ is a linear function of $n$.
$endgroup$
– Claude Leibovici
Jan 6 at 6:14
|
show 2 more comments
$begingroup$
The question is-
Write down the first five values of each of the following recursive sequences.
$S_0 = 1, S_n = (S_{n−1})^{2} + (S_{n−2})^{2} + · · · + (S_0)^{2}$ for all integers $n ⩾ 1$
I'm not sure if this is correct but this is what I did:
$S_1 = (1)^{2} = 1$
$S_2 = (1)^2 + (1)^2 = 2$
$S_3 = (2)^2 + (1)^2 + (1)^2 = 6$
$S_4 = (6)^2 + (2)^2 + (1)^2 + (1)^2 = 42$
$S_5 = (42)^2 + (6)^2 + (2)^2 + (1)^2 + (1)^2 = 1806$
proof-verification recursion
edited Jan 6 at 6:29
Shubham Johri
5,372718
asked Jan 6 at 5:42
llamaro25llamaro25
528
$endgroup$
The question is-
Write down the first five values of each of the following recursive sequences.
$S_0 = 1, S_n = (S_{n−1})^{2} + (S_{n−2})^{2} + · · · + (S_0)^{2}$ for all integers $n ⩾ 1$
I'm not sure if this is correct but this is what I did:
$S_1 = (1)^{2} = 1$
$S_2 = (1)^2 + (1)^2 = 2$
$S_3 = (2)^2 + (1)^2 + (1)^2 = 6$
$S_4 = (6)^2 + (2)^2 + (1)^2 + (1)^2 = 42$
$S_5 = (42)^2 + (6)^2 + (2)^2 + (1)^2 + (1)^2 = 1806$
proof-verification recursion
proof-verification recursion
edited Jan 6 at 6:29
Shubham Johri
5,372718
asked Jan 6 at 5:42
llamaro25llamaro25
528
edited Jan 6 at 6:29
Shubham Johri
5,372718
asked Jan 6 at 5:42
llamaro25llamaro25
528
edited Jan 6 at 6:29
Shubham Johri
5,372718
edited Jan 6 at 6:29
Shubham Johri
5,372718
edited Jan 6 at 6:29
Shubham Johri
5,372718
5,372718
asked Jan 6 at 5:42
llamaro25llamaro25
528
asked Jan 6 at 5:42
llamaro25llamaro25
528
asked Jan 6 at 5:42
llamaro25llamaro25
528
528
4
$begingroup$
Looks fine. What is the question, though? By the way, you can simplify to $S_n=S_{n-1}^2+S_{n-1}$.
$endgroup$
– SmileyCraft
Jan 6 at 5:44
$begingroup$
Thanks. This was the question: Write down the first five values of each of the following recursive sequences. S0 = 1, Sn = (Sn−1)^2 + (Sn−2)^2 + · · · + (S0)^2 for all integers n ⩾ 1
$endgroup$
– llamaro25
Jan 6 at 5:46
1
$begingroup$
@meromero25: Then you answered your own question? Any why is this interesting?
$endgroup$
– David G. Stork
Jan 6 at 5:50
$begingroup$
@DavidG.Stork I wasn't sure if the answer is correct
$endgroup$
– llamaro25
Jan 6 at 5:52
$begingroup$
Just for your curiosity : have a look at oeis.org/… What it seems is that $log(log(S_n))$ is a linear function of $n$.
$endgroup$
– Claude Leibovici
Jan 6 at 6:14
|
show 2 more comments
4
$begingroup$
Looks fine. What is the question, though? By the way, you can simplify to $S_n=S_{n-1}^2+S_{n-1}$.
$endgroup$
– SmileyCraft
Jan 6 at 5:44
$begingroup$
Thanks. This was the question: Write down the first five values of each of the following recursive sequences. S0 = 1, Sn = (Sn−1)^2 + (Sn−2)^2 + · · · + (S0)^2 for all integers n ⩾ 1
$endgroup$
– llamaro25
Jan 6 at 5:46
1
$begingroup$
@meromero25: Then you answered your own question? Any why is this interesting?
$endgroup$
– David G. Stork
Jan 6 at 5:50
$begingroup$
@DavidG.Stork I wasn't sure if the answer is correct
$endgroup$
– llamaro25
Jan 6 at 5:52
$begingroup$
Just for your curiosity : have a look at oeis.org/… What it seems is that $log(log(S_n))$ is a linear function of $n$.
$endgroup$
– Claude Leibovici
Jan 6 at 6:14
4
4
$begingroup$
Looks fine. What is the question, though? By the way, you can simplify to $S_n=S_{n-1}^2+S_{n-1}$.
$endgroup$
– SmileyCraft
Jan 6 at 5:44
$begingroup$
Looks fine. What is the question, though? By the way, you can simplify to $S_n=S_{n-1}^2+S_{n-1}$.
$endgroup$
– SmileyCraft
Jan 6 at 5:44
$begingroup$
Thanks. This was the question: Write down the first five values of each of the following recursive sequences. S0 = 1, Sn = (Sn−1)^2 + (Sn−2)^2 + · · · + (S0)^2 for all integers n ⩾ 1
$endgroup$
– llamaro25
Jan 6 at 5:46
$begingroup$
Thanks. This was the question: Write down the first five values of each of the following recursive sequences. S0 = 1, Sn = (Sn−1)^2 + (Sn−2)^2 + · · · + (S0)^2 for all integers n ⩾ 1
$endgroup$
– llamaro25
Jan 6 at 5:46
1
1
$begingroup$
@meromero25: Then you answered your own question? Any why is this interesting?
$endgroup$
– David G. Stork
Jan 6 at 5:50
$begingroup$
@meromero25: Then you answered your own question? Any why is this interesting?
$endgroup$
– David G. Stork
Jan 6 at 5:50
$begingroup$
@DavidG.Stork I wasn't sure if the answer is correct
$endgroup$
– llamaro25
Jan 6 at 5:52
$begingroup$
@DavidG.Stork I wasn't sure if the answer is correct
$endgroup$
– llamaro25
Jan 6 at 5:52
$begingroup$
Just for your curiosity : have a look at oeis.org/… What it seems is that $log(log(S_n))$ is a linear function of $n$.
$endgroup$
– Claude Leibovici
Jan 6 at 6:14
$begingroup$
Just for your curiosity : have a look at oeis.org/… What it seems is that $log(log(S_n))$ is a linear function of $n$.
$endgroup$
– Claude Leibovici
Jan 6 at 6:14
|
show 2 more comments
0
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4
$begingroup$
Looks fine. What is the question, though? By the way, you can simplify to $S_n=S_{n-1}^2+S_{n-1}$.
$endgroup$
– SmileyCraft
Jan 6 at 5:44
$begingroup$
Thanks. This was the question: Write down the first five values of each of the following recursive sequences. S0 = 1, Sn = (Sn−1)^2 + (Sn−2)^2 + · · · + (S0)^2 for all integers n ⩾ 1
$endgroup$
– llamaro25
Jan 6 at 5:46
1
$begingroup$
@meromero25: Then you answered your own question? Any why is this interesting?
$endgroup$
– David G. Stork
Jan 6 at 5:50
$begingroup$
@DavidG.Stork I wasn't sure if the answer is correct
$endgroup$
– llamaro25
Jan 6 at 5:52
$begingroup$
Just for your curiosity : have a look at oeis.org/… What it seems is that $log(log(S_n))$ is a linear function of $n$.
$endgroup$
– Claude Leibovici
Jan 6 at 6:14