Mixing
Recurrence relation of $1, 3, 3, 15, 5, 35, 7, 63, 9, 99, 11, 143, 13$
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
I found this sequence:
$$1,3,3,15,5,35,7,63,9,99,11,143,13...$$
I'm looking for its recurrence relation, and/or its closed form.
My take: it's easy to see that when $ngeq4$ and is even, $a_n=a_n-1cdot a_n+1$, and when $ngeq3$ and odd, then $a_n=n$. I'm having trouble to "combine" these observations into one recurrence relation.
Add: was able to write the recurrence relation as:
$$b_n=n^2-1+(-1)^n(b_n-1+(-1)^n-1(n-1))$$
For $b_1=1$, though the resulting closed form doesn't look elementary. The generating function, using a CAS, is:
$$fracx^5+x^4-6 x^3-3 x-1left(x^2-1right)^3$$
recurrence-relations integers
asked Jul 25 at 15:16
John Glenn
1,616123
 |Â
show 2 more comments
up vote
0
down vote
favorite
I found this sequence:
$$1,3,3,15,5,35,7,63,9,99,11,143,13...$$
I'm looking for its recurrence relation, and/or its closed form.
My take: it's easy to see that when $ngeq4$ and is even, $a_n=a_n-1cdot a_n+1$, and when $ngeq3$ and odd, then $a_n=n$. I'm having trouble to "combine" these observations into one recurrence relation.
Add: was able to write the recurrence relation as:
$$b_n=n^2-1+(-1)^n(b_n-1+(-1)^n-1(n-1))$$
For $b_1=1$, though the resulting closed form doesn't look elementary. The generating function, using a CAS, is:
$$fracx^5+x^4-6 x^3-3 x-1left(x^2-1right)^3$$
recurrence-relations integers
asked Jul 25 at 15:16
John Glenn
1,616123
math.meta.stackexchange.com/questions/28409/…
– Xander Henderson
Jul 25 at 15:18
If this sequence was in the OEIS, it would merit thedumbkeyword – it is not that interesting.
– Parcly Taxel
Jul 25 at 15:18
Where did this series come from?
– Brian Tung
Jul 25 at 15:49
2
A 'closed form' is simple: $a_n=n$, $n$ odd; $n^2-1$, $n$ even $$. This is a perfectly legitimate closed form. If you mean a single algebraic expression, that's possible too - just manipulate $(-1)^n$ appropriately - but there's generally nothing at all to be gained by such manipulations.
– Steven Stadnicki
Jul 25 at 15:50
@BrianTung I was looking at the first few terms of the summation in this question, the sequence above appears in the denominator of each term
– John Glenn
Jul 25 at 15:59
 |Â
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I found this sequence:
$$1,3,3,15,5,35,7,63,9,99,11,143,13...$$
I'm looking for its recurrence relation, and/or its closed form.
My take: it's easy to see that when $ngeq4$ and is even, $a_n=a_n-1cdot a_n+1$, and when $ngeq3$ and odd, then $a_n=n$. I'm having trouble to "combine" these observations into one recurrence relation.
Add: was able to write the recurrence relation as:
$$b_n=n^2-1+(-1)^n(b_n-1+(-1)^n-1(n-1))$$
For $b_1=1$, though the resulting closed form doesn't look elementary. The generating function, using a CAS, is:
$$fracx^5+x^4-6 x^3-3 x-1left(x^2-1right)^3$$
recurrence-relations integers
asked Jul 25 at 15:16
John Glenn
1,616123
I found this sequence:
$$1,3,3,15,5,35,7,63,9,99,11,143,13...$$
I'm looking for its recurrence relation, and/or its closed form.
My take: it's easy to see that when $ngeq4$ and is even, $a_n=a_n-1cdot a_n+1$, and when $ngeq3$ and odd, then $a_n=n$. I'm having trouble to "combine" these observations into one recurrence relation.
Add: was able to write the recurrence relation as:
$$b_n=n^2-1+(-1)^n(b_n-1+(-1)^n-1(n-1))$$
For $b_1=1$, though the resulting closed form doesn't look elementary. The generating function, using a CAS, is:
$$fracx^5+x^4-6 x^3-3 x-1left(x^2-1right)^3$$
recurrence-relations integers
asked Jul 25 at 15:16
John Glenn
1,616123
edited Jul 25 at 16:39
asked Jul 25 at 15:16
John Glenn
1,616123
asked Jul 25 at 15:16
John Glenn
1,616123
asked Jul 25 at 15:16
John Glenn
1,616123
1,616123
math.meta.stackexchange.com/questions/28409/…
– Xander Henderson
Jul 25 at 15:18
If this sequence was in the OEIS, it would merit thedumbkeyword – it is not that interesting.
– Parcly Taxel
Jul 25 at 15:18
Where did this series come from?
– Brian Tung
Jul 25 at 15:49
2
A 'closed form' is simple: $a_n=n$, $n$ odd; $n^2-1$, $n$ even $$. This is a perfectly legitimate closed form. If you mean a single algebraic expression, that's possible too - just manipulate $(-1)^n$ appropriately - but there's generally nothing at all to be gained by such manipulations.
– Steven Stadnicki
Jul 25 at 15:50
@BrianTung I was looking at the first few terms of the summation in this question, the sequence above appears in the denominator of each term
– John Glenn
Jul 25 at 15:59
 |Â
show 2 more comments
math.meta.stackexchange.com/questions/28409/…
– Xander Henderson
Jul 25 at 15:18
If this sequence was in the OEIS, it would merit thedumbkeyword – it is not that interesting.
– Parcly Taxel
Jul 25 at 15:18
Where did this series come from?
– Brian Tung
Jul 25 at 15:49
2
A 'closed form' is simple: $a_n=n$, $n$ odd; $n^2-1$, $n$ even $$. This is a perfectly legitimate closed form. If you mean a single algebraic expression, that's possible too - just manipulate $(-1)^n$ appropriately - but there's generally nothing at all to be gained by such manipulations.
– Steven Stadnicki
Jul 25 at 15:50
@BrianTung I was looking at the first few terms of the summation in this question, the sequence above appears in the denominator of each term
– John Glenn
Jul 25 at 15:59
math.meta.stackexchange.com/questions/28409/…
– Xander Henderson
Jul 25 at 15:18
math.meta.stackexchange.com/questions/28409/…
– Xander Henderson
Jul 25 at 15:18
If this sequence was in the OEIS, it would merit the
dumb keyword – it is not that interesting.– Parcly Taxel
Jul 25 at 15:18
If this sequence was in the OEIS, it would merit the
dumb keyword – it is not that interesting.– Parcly Taxel
Jul 25 at 15:18
Where did this series come from?
– Brian Tung
Jul 25 at 15:49
Where did this series come from?
– Brian Tung
Jul 25 at 15:49
2
2
A 'closed form' is simple: $a_n=n$, $n$ odd; $n^2-1$, $n$ even $$. This is a perfectly legitimate closed form. If you mean a single algebraic expression, that's possible too - just manipulate $(-1)^n$ appropriately - but there's generally nothing at all to be gained by such manipulations.
– Steven Stadnicki
Jul 25 at 15:50
A 'closed form' is simple: $a_n=n$, $n$ odd; $n^2-1$, $n$ even $$. This is a perfectly legitimate closed form. If you mean a single algebraic expression, that's possible too - just manipulate $(-1)^n$ appropriately - but there's generally nothing at all to be gained by such manipulations.
– Steven Stadnicki
Jul 25 at 15:50
@BrianTung I was looking at the first few terms of the summation in this question, the sequence above appears in the denominator of each term
– John Glenn
Jul 25 at 15:59
@BrianTung I was looking at the first few terms of the summation in this question, the sequence above appears in the denominator of each term
– John Glenn
Jul 25 at 15:59
 |Â
show 2 more comments
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2862517%2frecurrence-relation-of-1-3-3-15-5-35-7-63-9-99-11-143-13%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
math.meta.stackexchange.com/questions/28409/…
– Xander Henderson
Jul 25 at 15:18
If this sequence was in the OEIS, it would merit the
dumbkeyword – it is not that interesting.– Parcly Taxel
Jul 25 at 15:18
Where did this series come from?
– Brian Tung
Jul 25 at 15:49
2
A 'closed form' is simple: $a_n=n$, $n$ odd; $n^2-1$, $n$ even $$. This is a perfectly legitimate closed form. If you mean a single algebraic expression, that's possible too - just manipulate $(-1)^n$ appropriately - but there's generally nothing at all to be gained by such manipulations.
– Steven Stadnicki
Jul 25 at 15:50
@BrianTung I was looking at the first few terms of the summation in this question, the sequence above appears in the denominator of each term
– John Glenn
Jul 25 at 15:59