Find a combinatorial interpretation of $ x_n = x_{n - 2} + x_{n - 3}$
1
$begingroup$
Let $ x_0 = 3, x_1 = 0, x_2 = 2$ and: $$ x_n = x_{n - 2} + x_{n - 3}$$
How to find a combinatorial interpretation of this equation?
One idea was to find a number of different tiles of a circle divided on $n$ unit arcs with arcs of lenght $2$ and $3$, but this obviously fails.
combinatorics
asked Jan 18 at 13:01
Maria MazurMaria Mazur
50.3k1361126
$endgroup$
add a comment |
1
$begingroup$
Let $ x_0 = 3, x_1 = 0, x_2 = 2$ and: $$ x_n = x_{n - 2} + x_{n - 3}$$
How to find a combinatorial interpretation of this equation?
One idea was to find a number of different tiles of a circle divided on $n$ unit arcs with arcs of lenght $2$ and $3$, but this obviously fails.
combinatorics
asked Jan 18 at 13:01
Maria MazurMaria Mazur
50.3k1361126
$endgroup$
2
$begingroup$
This is A001608 and there are some suggestions given there (though nothing I'd say was terribly compelling).
$endgroup$
– lulu
Jan 18 at 13:07
2
$begingroup$
More references and a not particularly funny cartoon are here.
$endgroup$
– lulu
Jan 18 at 13:08
2
$begingroup$
And here (though no cartoon).
$endgroup$
– lulu
Jan 18 at 13:17
1
$begingroup$
I haven't a combinatorial interpretation. May I ask you why you are interested by this sequence ? Because, with different initial values, it is called the Padovan sequence oeis.org/A000931 and has a geometric interpretation. See page 70 of this very nice on-line book m-hikari.com/mccartin-2.pdf
$endgroup$
– Jean Marie
Mar 11 at 4:19
$begingroup$
Yes, of course. For this sequence we have $pmid x_p$ if $p$ is prime. I want to find a combinatorial argument for this. And thanks you for pointig me to this book. It is marvelous!
$endgroup$
– Maria Mazur
Mar 11 at 18:16
add a comment |
1
1
1
$begingroup$
Let $ x_0 = 3, x_1 = 0, x_2 = 2$ and: $$ x_n = x_{n - 2} + x_{n - 3}$$
How to find a combinatorial interpretation of this equation?
One idea was to find a number of different tiles of a circle divided on $n$ unit arcs with arcs of lenght $2$ and $3$, but this obviously fails.
combinatorics
asked Jan 18 at 13:01
Maria MazurMaria Mazur
50.3k1361126
$endgroup$
Let $ x_0 = 3, x_1 = 0, x_2 = 2$ and: $$ x_n = x_{n - 2} + x_{n - 3}$$
How to find a combinatorial interpretation of this equation?
One idea was to find a number of different tiles of a circle divided on $n$ unit arcs with arcs of lenght $2$ and $3$, but this obviously fails.
combinatorics
combinatorics
asked Jan 18 at 13:01
Maria MazurMaria Mazur
50.3k1361126
asked Jan 18 at 13:01
Maria MazurMaria Mazur
50.3k1361126
asked Jan 18 at 13:01
Maria MazurMaria Mazur
50.3k1361126
asked Jan 18 at 13:01
Maria MazurMaria Mazur
50.3k1361126
asked Jan 18 at 13:01
Maria MazurMaria Mazur
50.3k1361126
50.3k1361126
2
$begingroup$
This is A001608 and there are some suggestions given there (though nothing I'd say was terribly compelling).
$endgroup$
– lulu
Jan 18 at 13:07
2
$begingroup$
More references and a not particularly funny cartoon are here.
$endgroup$
– lulu
Jan 18 at 13:08
2
$begingroup$
And here (though no cartoon).
$endgroup$
– lulu
Jan 18 at 13:17
1
$begingroup$
I haven't a combinatorial interpretation. May I ask you why you are interested by this sequence ? Because, with different initial values, it is called the Padovan sequence oeis.org/A000931 and has a geometric interpretation. See page 70 of this very nice on-line book m-hikari.com/mccartin-2.pdf
$endgroup$
– Jean Marie
Mar 11 at 4:19
$begingroup$
Yes, of course. For this sequence we have $pmid x_p$ if $p$ is prime. I want to find a combinatorial argument for this. And thanks you for pointig me to this book. It is marvelous!
$endgroup$
– Maria Mazur
Mar 11 at 18:16
add a comment |
2
$begingroup$
This is A001608 and there are some suggestions given there (though nothing I'd say was terribly compelling).
$endgroup$
– lulu
Jan 18 at 13:07
2
$begingroup$
More references and a not particularly funny cartoon are here.
$endgroup$
– lulu
Jan 18 at 13:08
2
$begingroup$
And here (though no cartoon).
$endgroup$
– lulu
Jan 18 at 13:17
1
$begingroup$
I haven't a combinatorial interpretation. May I ask you why you are interested by this sequence ? Because, with different initial values, it is called the Padovan sequence oeis.org/A000931 and has a geometric interpretation. See page 70 of this very nice on-line book m-hikari.com/mccartin-2.pdf
$endgroup$
– Jean Marie
Mar 11 at 4:19
$begingroup$
Yes, of course. For this sequence we have $pmid x_p$ if $p$ is prime. I want to find a combinatorial argument for this. And thanks you for pointig me to this book. It is marvelous!
$endgroup$
– Maria Mazur
Mar 11 at 18:16
2
2
$begingroup$
This is A001608 and there are some suggestions given there (though nothing I'd say was terribly compelling).
$endgroup$
– lulu
Jan 18 at 13:07
$begingroup$
This is A001608 and there are some suggestions given there (though nothing I'd say was terribly compelling).
$endgroup$
– lulu
Jan 18 at 13:07
2
2
$begingroup$
More references and a not particularly funny cartoon are here.
$endgroup$
– lulu
Jan 18 at 13:08
$begingroup$
More references and a not particularly funny cartoon are here.
$endgroup$
– lulu
Jan 18 at 13:08
2
2
$begingroup$
And here (though no cartoon).
$endgroup$
– lulu
Jan 18 at 13:17
$begingroup$
And here (though no cartoon).
$endgroup$
– lulu
Jan 18 at 13:17
1
1
$begingroup$
I haven't a combinatorial interpretation. May I ask you why you are interested by this sequence ? Because, with different initial values, it is called the Padovan sequence oeis.org/A000931 and has a geometric interpretation. See page 70 of this very nice on-line book m-hikari.com/mccartin-2.pdf
$endgroup$
– Jean Marie
Mar 11 at 4:19
$begingroup$
I haven't a combinatorial interpretation. May I ask you why you are interested by this sequence ? Because, with different initial values, it is called the Padovan sequence oeis.org/A000931 and has a geometric interpretation. See page 70 of this very nice on-line book m-hikari.com/mccartin-2.pdf
$endgroup$
– Jean Marie
Mar 11 at 4:19
$begingroup$
Yes, of course. For this sequence we have $pmid x_p$ if $p$ is prime. I want to find a combinatorial argument for this. And thanks you for pointig me to this book. It is marvelous!
$endgroup$
– Maria Mazur
Mar 11 at 18:16
$begingroup$
Yes, of course. For this sequence we have $pmid x_p$ if $p$ is prime. I want to find a combinatorial argument for this. And thanks you for pointig me to this book. It is marvelous!
$endgroup$
– Maria Mazur
Mar 11 at 18:16
add a comment |
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2
$begingroup$
This is A001608 and there are some suggestions given there (though nothing I'd say was terribly compelling).
$endgroup$
– lulu
Jan 18 at 13:07
2
$begingroup$
More references and a not particularly funny cartoon are here.
$endgroup$
– lulu
Jan 18 at 13:08
2
$begingroup$
And here (though no cartoon).
$endgroup$
– lulu
Jan 18 at 13:17
1
$begingroup$
I haven't a combinatorial interpretation. May I ask you why you are interested by this sequence ? Because, with different initial values, it is called the Padovan sequence oeis.org/A000931 and has a geometric interpretation. See page 70 of this very nice on-line book m-hikari.com/mccartin-2.pdf
$endgroup$
– Jean Marie
Mar 11 at 4:19
$begingroup$
Yes, of course. For this sequence we have $pmid x_p$ if $p$ is prime. I want to find a combinatorial argument for this. And thanks you for pointig me to this book. It is marvelous!
$endgroup$
– Maria Mazur
Mar 11 at 18:16