A085223 -id:A085223 - cp's OEIS frontend

A072795 A014486-indices of the plane binary trees AND plane general trees whose left subtree is just a stick: \. thus corresponding to the parenthesizations whose first element (of the top-level list) is an empty parenthesization: ().

1, 2, 4, 5, 9, 10, 11, 12, 13, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 197, 198, 199

Offset: 0

Antti Karttunen Jun 12 2002

nonn

This sequence is induced by the 'flipped form' of the function 'list': (define (flippedlist x) (cons '() x)) when it acts on symbolless S-expressions encoded by A014486/A063171.

Gives in A063171 positions of the terms which begin with digits 10...

Column 0 of A072764, row 0 of A072766, column 1 of A085201. Complement: A081291. Cf. A085223.

Range[0, Length[#]-1] + CatalanNumber[#] & [Flatten[Array[Table[#, CatalanNumber[#]] &, 7, 0]]] (* Paolo Xausa, Mar 01 2024 *)

a(n) = n + A000108(A072643(n)) = A069770(A057548(n)) = A080300(A083937(n))

A085201 Array A(x,y): Position of the concatenation of binary strings A014486(x) & A014486(y) in A014486, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 9, 5, 4, 5, 9, 14, 10, 9, 5, 6, 11, 23, 15, 23, 10, 6, 7, 14, 28, 24, 37, 24, 11, 7, 8, 16, 37, 29, 65, 38, 25, 12, 8, 9, 19, 42, 38, 79, 66, 39, 26, 13, 9, 10, 23, 51, 43, 107, 80, 67, 40, 27, 23, 10, 11, 25, 65, 52, 121, 108, 81, 68, 41, 65, 24, 11

Offset: 0

Antti Karttunen, Jun 23 2003

This table is induced by the 2-ary form of the list-function 'append' present in the programming languages like Lisp, Scheme and Prolog.

Index entries for the sequences induced by list functions of Lisp

Transpose: A085202. Variant: A085203. Row 1: A085223, Column 1: A072795.

Cf. also A014486, A080300, A025581, A002262.

a(0, y)=y, a(x, y) = A072764bi(A072771(x), a(A072772(x), y))

a(x, y) = A080300(A085207bi(A014486(x), A014486(y))) = A085200(A085215bi(A071155(y), A071155(x)))

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)).

A085178 Array A(x,y) giving the position of the y-th x in A080237 listed by rising antidiagonals.

Original entry on oeis.org

1, 3, 2, 8, 5, 4, 22, 13, 7, 6, 64, 36, 18, 10, 9, 196, 106, 50, 21, 12, 11, 625, 328, 148, 59, 27, 15, 14, 2055, 1054, 460, 176, 63, 32, 17, 16, 6917, 3485, 1483, 550, 190, 78, 35, 20, 19, 23713, 11779, 4915, 1780, 598, 195, 92, 41, 24, 23, 82499, 40509, 16641, 5916

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.

Index entries for sequences that are permutations of the natural numbers

Transpose: A085176.
Inverse permutation: A085179.
The first row: A014138, the first column: A085223.
Variant: A085180.

A085196 Difference of row 1 and column 1 of the array A085201.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 3, 4, 6, 0, 1, 3, 4, 6, 9, 10, 12, 13, 15, 18, 19, 21, 24, 0, 1, 3, 4, 6, 9, 10, 12, 13, 15, 18, 19, 21, 24, 28, 29, 31, 32, 34, 37, 38, 40, 41, 43, 46, 47, 49, 52, 56, 57, 59, 60, 62, 65, 66, 68, 71, 75, 76, 78, 81, 85, 0, 1, 3, 4, 6, 9, 10, 12, 13, 15, 18, 19, 21

Offset: 0

Antti Karttunen, Jun 14 2003

nonn

The repeating part: A080336 = A085196 o A081291. Cf. A085197.

a(n) = A085223(n) - A072795(n).

A085224 A014486-encodings of the plane general trees whose rightmost subtree (branching from the root) is just a stick: /.

Original entry on oeis.org

2, 10, 42, 50, 170, 178, 202, 210, 226, 682, 690, 714, 722, 738, 810, 818, 842, 850, 866, 906, 914, 930, 962, 2730, 2738, 2762, 2770, 2786, 2858, 2866, 2890, 2898, 2914, 2954, 2962, 2978, 3010, 3242, 3250, 3274, 3282, 3298, 3370, 3378, 3402, 3410, 3426, 3466

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

Index entries for encodings of plane rooted trees

a(n) = (4*A014486(n)) + 2 = A014486(A085223(n))

A130374 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the last element in place if the length (A057515(n)) is odd.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 14, 16, 19, 11, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 24, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 45, 46, 32, 48, 49, 50, 33, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 66, 70, 72, 75, 67, 71

Offset: 0

Antti Karttunen, Jun 05 2007

This self-inverse automorphism permutes the top level of a list of even length (1 2 3 4 ... 2n-1 2n) as (2 1 4 3 ... 2n 2n-1), and when applied to a list of odd length (1 2 3 4 ... 2n-1 2n 2n+1), permutes it as (2 1 4 3 ... 2n 2n-1 2n+1).

Antti Karttunen, Table of n, a(n) for n = 0..2055
Index entries for signature-permutations of Catalan automorphisms

Cf. a(n) = A057508(A130373(A057508(n))) = A057164(A130373(A057164(n))) = A127285(A127288(n)) = A127287(A127286(n)). Also a(A085223(n)) = A130370(A122282(A130369(A085223(n)))) holds for all n>=0. The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A073193 and A073192.

cp's OEIS Frontend

A072795 A014486-indices of the plane binary trees AND plane general trees whose left subtree is just a stick: \. thus corresponding to the parenthesizations whose first element (of the top-level list) is an empty parenthesization: ().

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A085201 Array A(x,y): Position of the concatenation of binary strings A014486(x) & A014486(y) in A014486, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

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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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A085178 Array A(x,y) giving the position of the y-th x in A080237 listed by rising antidiagonals.

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A085196 Difference of row 1 and column 1 of the array A085201.

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A085224 A014486-encodings of the plane general trees whose rightmost subtree (branching from the root) is just a stick: /.

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A130374 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the last element in place if the length (A057515(n)) is odd.

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