Do any integers occur in both sequences?

[Leroy Quet]

says (near top):
  Number of sequences of length n-1 consisting of positive integers
  such that the opening and ending elements are 1 or 2, and the
  absolute difference between any 2 consecutive elements is 0 or 1.
 A024537 ,1, 2, 4, 9, 21, 50, 120, 289, 697, 1682 (=?= A018905)
     [1,] 1, 2, 4, 9, 21, 51, 127, 323,
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Date: Tue, 31 Jul 2007 10:50:49 -0700
From: "Max Alekseyev" <maxale at gmail.com>
To: "Joerg Arndt" <arndt at jjj.de>
Subject: Re: possibly wrong comment with Motzkin numbers
Cc: seqfan at ext.jussieu.fr
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On 7/31/07, Joerg Arndt <arndt at jjj.de> wrote:
> A comment with seq A001006
> http://www.research.att.com/~njas/sequences/A001006
> says (near top):
>   Number of sequences of length n-1 consisting of positive integers
>   such that the opening and ending elements are 1 or 2, and the
>   absolute difference between any 2 consecutive elements is 0 or 1.
>
> This seems to be wrong, I get the counts
>  A024537 ,1, 2, 4, 9, 21, 50, 120, 289, 697, 1682 (=?= A018905)
>
> Note the starts math, Motzkins start as
>      [1,] 1, 2, 4, 9, 21, 51, 127, 323,

With a simple recurrence formula implementation in PARI/GP, I confirm
that the comment in A001006 is correct:

{ a(n,l) = if(n==1,(l==1)||(l==2),if(l<=0,0,a(n-1,l-1)+a(n-1,l)+a(n-1,l+1))) }

{ f(n) = a(n,1)+a(n,2) }

? vector(10,n,f(n))
%1 = [2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798]

Here a(n,l) counts the number of sequences of positive integers of
length n, starting with 1 or 2 and ending with l.

Max