

A076050


Limiting sequence if we start with 2 and successively replace n with 2,3,4,...,n,n+1.


6



2, 3, 2, 3, 4, 2, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 2, 3, 4, 2, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 4, 5, 6, 2, 3, 2, 3, 4, 2, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 2, 3, 4, 2, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 4, 5, 6, 2, 3, 2, 3, 4, 2, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 2, 3, 4, 2, 3
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OFFSET

1,1


COMMENTS

We get 2, 23, 23234, 23234232342345 and so on. The lengths are 1,2,5,14,42,... which are the Catalan numbers: A000108. The sums of the numbers in these strings are also the Catalan numbers.
In A071159 the ndigit terms follow the 2, 3, 2, 3, 4, ... rule: the number of terms in which the first n1 digits are the same is 2, 3, 2, 3, 4, ... and the last digits of the terms are 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, ..., A007001. For example, 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1221, 1222, 1223, 1231, 1232, 1233, 1234.
a(A000108(n)) = n+1 and a(m) < n+1 for m < A000108(n).  Reinhard Zumkeller, Feb 17 2012
Let (T(1) < T(2) < ... < T(A000108(m))) denote the sequence of Young tableaux of shape (2^m) ordered lexicographically with respect to their columns, and let f(T(i), T(j)) denote the first label of disagreement among T(i) and T(j). Then, empirically, the reverse of the list (f(T(1), T(2)), f(T(1), T(3)), ..., f(T(1), T(A000108(m)))) agrees with the first A000108(m)  1 terms in this sequence, for all m > 1, as illustrated in the below example.  John M. Campbell, Sep 07 2018


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Italo J. Dejter, The role of restricted growth strings in the two middle levels of the Boolean lattice B_(2k+1), University of Puerto Rico, 2018.


EXAMPLE

From John M. Campbell, Sep 07 2018: (Start)
There are A000108(3) = 5 Young tableaux of shape (2^3) = (2, 2, 2), which are listed below lexicographically.
[3 6] [4 6] [4 6] [5 6] [5 6]
[2 5] < [2 5] < [3 5] < [2 4] < [3 4]
[1 4] [1 3] [1 2] [1 3] [1 2]
As above, let (T(1), T(2), ..., T(5)) denote this list. The first label of disagreement between T(1) and T(5) is 2; that between T(1) and T(4) is 3; that between T(1) and T(3) is 2; that between T(1) and T(2) is 3. The sequence (2, 3, 2, 3) agrees with the first 4 terms in this sequence. If we repeat this process using Young tableaux of shape (2^4), we obtain the sequence (2, 3, 2, 3, 4, 2, 3, 2, 3, 4, 2, 3, 4). (End)


PROG

(PARI) a(n)=local(v, w); if(n<1, 0, v=[1]; while(#v<n, w=[]; for(i=1, #v, w=concat(w, vector(v[i]+1, j, j))); v=w); 1+v[n])
(Haskell)
a076050 n = a076050_list !! (n1)
a076050_list = 2 : f [2] where
f xs = (drop (length xs) xs') ++ (f xs') where
xs' = concatMap ((enumFromTo 2) . (+ 1)) xs
 Reinhard Zumkeller, Feb 17 2012


CROSSREFS

Cf. A000108, A071159. a(n)=A007001(n)+1.
Sequence in context: A293519 A237582 A097352 * A130799 A243519 A278102
Adjacent sequences: A076047 A076048 A076049 * A076051 A076052 A076053


KEYWORD

easy,nonn,nice


AUTHOR

Miklos Kristof, Oct 30 2002


STATUS

approved



