login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A080237 Start with 1 and apply the process: k-th run is 1, 2, 3, ..., a(k-1)+1. 12
1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 1, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Also a triangle collected from the Catalan generating tree, with row n containing A000108(n) terms:

1;

1,2;

1,2,1,2,3;

1,2,1,2,3,1,2,1,2,3,1,2,3,4;

and ending with n. Rows converge towards A007001, the "last" row.

LINKS

Reinhard Zumkeller, >Rows n = 1..10 of triangle, flattened

C. Banderier, A. Denise, P. Flajolet, M. Bousquet-Mélou et al., Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.

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

R. P. Stanley, Catalan addendum. See the interpretation (www, "Vertices of height n-1 of the tree T ...").

FORMULA

It seems that Sum_{k=1..n} a(k) = C*n*log(log(n)) + O(n*log(log(n))) with C = 0.6....

a(n) = A007814(A014486(n)) (i.e., number of trailing zeros in A063171(n)).

EXAMPLE

Sequence begins: 1,(1,2),(1,2),(1,2,3), ... where runs are between 2 parentheses. 5th run is (1,2) since a(4)=1 and sequence continues: 1,1,2,1,2,1,2,3,1,2....

G.f. = x + x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 + 2*x^7 + 3*x^8 + x^9 + 2*x^10 + ...

MATHEMATICA

run[1] = {1}; run[k_] := run[k] = Range[ Flatten[ Table[run[j], {j, 1, k-1}]][[k-1]] + 1]; Table[run[k], {k, 1, 29}] // Flatten (* Jean-François Alcover, Sep 12 2012 *)

NestList[ Flatten[# /. # -> Range[# + 1]] &, {1}, 5] // Flatten (* Robert G. Wilson v, Jun 24 2014 *)

PROG

(PARI) {a(n) = my(v, i, j, k); if( n<1, 0, v=vector(n); for(m=1, n, v[m]=k++; if( k>j, j=v[i++]; k=0)); v[n])}; /* Michael Somos, Jun 24 2014 */

(Haskell)

a080237 n k = a080237_tabf !! (n-1) !! (k-1)

a080237_row n = a080237_tabf !! (n-1)

a080237_tabf = [1] : f a080237_tabf where

   f [[]] =[]

   f (xs:xss) = concatMap (enumFromTo 1 . (+ 1)) xs : f xss

a080237_list = concat a080237_tabf

-- Reinhard Zumkeller, Jun 01 2015

CROSSREFS

Cf. A000002, A007001. Positions of ones: A085223. The first occurrence of each n is at A014138(n). See A085178.

Sequence in context: A133780 A270808 A290532 * A136109 A105265 A193360

Adjacent sequences:  A080234 A080235 A080236 * A080238 A080239 A080240

KEYWORD

nonn,tabf

AUTHOR

Benoit Cloitre, Mar 18 2003

EXTENSIONS

Additional comments from Antti Karttunen, Jun 17 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 03:00 EST 2019. Contains 329836 sequences. (Running on oeis4.)