Combinatorics - Building strings with symbols from two alphabets Given two alphabets $A$ and $B$ each of cardinality $L$, how many strings of length $2L$ can I build such that
a) No symbol is repeated and
b) No more than two symbols of A appear consecutively.
 A: Hints:  you are going to use all the letters.  First how many orders are there of all the symbols from $A$ alone?  How many orders for all the symbols of $B$ alone?  These orders can be paired arbitrarily.  Then we need to choose which of the  $2L$ slots get letters from $A$.  How many ways are there to select those?  You could give all the odd numbered slots to $A$, or ...
Added:  I was thinking no two A's in a row, not no more than two.  The number of configurations starts $2,6,16,45,126,357,1016,\ldots $The series appears to be given in OEIS, but I don't see how to derive it.  The last entry in comments says this is the series we want.  The way I got the numbers was to define $C(n,m)$ as the number of series of $n A$'s and $m B$'s that don't have three or more successive $A$'s and don't end in $A$, $D(n,m)$ as the number of series of $n A$'s and $m B$'s that don't have three or more successive $A$'s and end in one $A$, $E(n,m)$ as the number of series of $n A$'s and $m B$'s that don't have three or more successive $A$'s and end in two $A$'s, $F(n,m)$ as the total number of series don't have three or more successive $A$'s.  $C(0,0)=1, D(0,0)=0, E(0,0)=0, C(n,m)=C(n,m-1)+D(n,m-1)+E(n,m-1)$
$ D(n,m)=C(n-1,m), E(n,m)=D(n-1,m), F(n,m)=C(n,m)+d(n,m)+E(n,m)$
I just put it all in a spreadsheet, found the numbers above, and looked up in OEIS.
