A085178 -id:A085178 - cp's OEIS frontend

A085176 Transpose of A085178.

1, 2, 3, 4, 5, 8, 6, 7, 13, 22, 9, 10, 18, 36, 64, 11, 12, 21, 50, 106, 196, 14, 15, 27, 59, 148, 328, 625, 16, 17, 32, 63, 176, 460, 1054, 2055, 19, 20, 35, 78, 190, 550, 1483, 3485, 6917, 23, 24, 41, 92, 195, 598, 1780, 4915, 11779, 23713, 25, 26, 46, 101, 238

Offset: 1

Antti Karttunen, Jun 18 2003

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A085177.

A085179 Inverse permutation to A085178.

Original entry on oeis.org

1, 3, 2, 6, 5, 10, 9, 4, 15, 14, 21, 20, 8, 28, 27, 36, 35, 13, 45, 44, 19, 7, 55, 54, 66, 65, 26, 78, 77

Offset: 1

Antti Karttunen, Jun 18 2003

Index entries for sequences that are permutations of the natural numbers

A080237 Start with 1 and apply the process: k-th run is 1, 2, 3, ..., a(k-1)+1.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Mar 18 2003

Also a triangle collected from the Catalan generating tree, with row n containing A000108(n) terms and ending with n. Rows converge towards A007001, the "last" row. - Antti Karttunen, Jun 17 2003

			As an irregular triangle:
  1;
  1,2;
  1,2,1,2,3;
  1,2,1,2,3,1,2,1,2,3,1,2,3,4;
  ...
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 + ...

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.
Antti 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 ...").

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

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

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 *)

{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 */

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)).

A085180 Array A(x,y) giving the position of the y-th x in A007001 listed by rising antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 14, 10, 7, 8, 42, 28, 13, 9, 11, 132, 84, 37, 19, 12, 15, 429, 264, 112, 41, 24, 16, 17, 1430, 858, 354, 126, 56, 27, 18, 20, 4862, 2860, 1155, 402, 131, 70, 33, 21, 22, 16796, 9724, 3861, 1320, 422, 174, 79, 36, 23, 25, 58786, 33592, 13156, 4433

Offset: 1

Antti Karttunen, Jun 18 2003

Read by upwards antidiagonals as A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), etc.

			In A007001 each n occurs for the first time at position Cat(n), thus A(x,1) (the first row) is A000108.

Index entries for sequences that are permutations of the natural numbers

Variant: A085178.
Inverse permutation: A085181.
First row: A000108, second row: A068875 apart from initial terms, first column: A085197, central diagonal: A001453.

cp's OEIS Frontend

A085176 Transpose of A085178.

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A085179 Inverse permutation to A085178.

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A080237 Start with 1 and apply the process: k-th run is 1, 2, 3, ..., a(k-1)+1.

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A085180 Array A(x,y) giving the position of the y-th x in A007001 listed by rising antidiagonals.

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