

A080237


Start with 1 and apply the process: kth run is 1, 2, 3, ..., a(k1)+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
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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. BousquetMélou et al., Generating Functions for Generating Trees, Discrete Mathematics 246(13), March 2002, pp. 2955.
A. Karttunen, Notes concerning A080237tree and related sequences.
R. P. Stanley, Catalan addendum. See the interpretation (www, "Vertices of height n1 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, k1}]][[k1]] + 1]; Table[run[k], {k, 1, 29}] // Flatten (* JeanFrançois 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 !! (n1) !! (k1)
a080237_row n = a080237_tabf !! (n1)
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



