A007001 -id:A007001 - cp's OEIS frontend

A076050 Limiting sequence if we start with 2 and successively replace n with 2,3,4,...,n,n+1.

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

Offset: 1

Miklos Kristof, Oct 30 2002

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 n-digit terms follow the 2, 3, 2, 3, 4, ... rule: the number of terms in which the first n-1 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

			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)

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.
Italo J. Dejter, Reinterpreting Mütze's Theorem via Natural Enumeration of Ordered Rooted Trees, arXiv:1911.02100 [math.CO], 2019.

Cf. A000108, A071159, A007001.

a076050 n = a076050_list !! (n-1)
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

Nest[Flatten[Map[Range[2, #+1] &, #]] &, {2}, 5] (* Paolo Xausa, Mar 04 2024 *)

a(n)=local(v,w); if(n<1,0,v=[1]; while(#v

a(n) = A007001(n) + 1.

A085193 Repeating part of A085192.

Original entry on oeis.org

2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 58, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 26, 2, 6, 2, 4, 10, 2, 4, 8, 18, 2, 4, 8, 16, 202, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 58, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 26, 2, 6, 2, 4, 10, 2, 4, 8, 18, 2, 4, 8, 16, 74, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10

Offset: 0

Antti Karttunen, Jun 14 2003

nonn

Antti Karttunen, Notes concerning A080237-tree and related sequences.

Same sequence divided by 2: A085194. Cf. A085190.

a(n) = A085192(A081291(n+1)-1).

a(n) = 4*A085193(A085182(n+1)-1) + 2 - (2^A007001(n+1)) if A007001(n+2)==1, otherwise 2^A007001(n+1).

A133914 Row sums of triangle A133913.

Original entry on oeis.org

1, 3, 5, 11, 23, 44, 89, 177, 355, 711, 1420, 2841, 5683, 11367, 22731, 45463, 90925, 181851, 363703, 727404, 1454809, 2909617, 5819235, 11638471, 23276940, 46553881, 93107763, 186215527, 372431051, 744862103, 1489724205, 2979448411, 5958896823, 11917793644

Offset: 1

Gary W. Adamson, Sep 28 2007

nonn

			a(4) = 11 = sum of row 4 terms of triangle A133913: (1 + 4 + 4 + 2).
a(4) = 11 = 2*a(3) + A133912(4) = 2*5 + 1, where A133912 = (1, 1, -1, 1, 1, -2, 1, -1, ...), first finite difference row of A007001.

Cf. A007001, A133912, A133913.

a(n) = 2 * a(n-1) + A133912(n). - Jack W Grahl, Oct 11 2022

More terms from Jack W Grahl, Oct 11 2022

A229830 Trajectory of 1 under the morphism 1 -> 12, 2 -> 1232, 3 -> 123432, 4 -> 12345432, etc.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1

Offset: 1

Franklin T. Adams-Watters, Dec 08 2013

nonn

It appears that records occur at A000111. [This is false. - Rémy Sigrist, Nov 15 2018]
The n-th local maximum equals a(n) + 1. - Rémy Sigrist, Nov 13 2018
Records at {1, 2, 5, 16, 61, 258, 1161, 5440, 26233, 129282, 648141, 3294864, ...}, which seems to match A104858. - Michael De Vlieger, Nov 13 2018

Rémy Sigrist, Table of n, a(n) for n = 1..8558 (7 iterations)
Index entries for sequences that are fixed points of mappings

Cf. A007001, A000111.

With[{n = 5}, Take[#, LengthWhile[#, # <= n &] + 1] &@ Nest[Flatten[# /. Array[Last@ # - 1 -> Most[#~Join~Reverse@ Most@ #] &@ Range@ # &, Max@ #, 2]] &, {1}, n]] (* Michael De Vlieger, Nov 13 2018 *)

my(a = vector(87), n=0); a[1]=1; for (p=1, oo, my(h=1+a[p]); for (v=1, h, a[n++]=v; print1 (v ", "); if (n==#a, break (2))); forstep (v=h-1, 2, -1, a[n++]=v; print1 (v ", "); if (n==#a, break (2)))) \\ Rémy Sigrist, Nov 13 2018

Data corrected by Rémy Sigrist, Nov 13 2018

A162696 Trajectory of 1 under morphism taking n to sorted divisors of n+1.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 3, 1, 5, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 2, 3, 6, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 3, 1, 5, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 3, 1, 5, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 4, 1, 7, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2

Offset: 1

Franklin T. Adams-Watters, Jul 10 2009

nonn

1 -> 1,2; 2->1,3; 3->1,2,4; ...

			1 -> 1,2 -> 1,2,1,3 -> 1,2,1,3,1,2,1,2,4 -> ...

Index entries for sequences that are fixed points of mappings

Cf. A007001, A001511, A027750, A112682, A117160.

v=[1,2];for(i=2,60,v=concat(v,divisors(v[i]+1)));v

a(A117160(k+1)) = k (this is the first occurrence of k in the sequence). - Rémy Sigrist, Jan 14 2023

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A076050 Limiting sequence if we start with 2 and successively replace n with 2,3,4,...,n,n+1.

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A085193 Repeating part of A085192.

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A133914 Row sums of triangle A133913.

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A229830 Trajectory of 1 under the morphism 1 -> 12, 2 -> 1232, 3 -> 123432, 4 -> 12345432, etc.

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A162696 Trajectory of 1 under morphism taking n to sorted divisors of n+1.

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