A081289 -id:A081289 - cp's OEIS frontend

A081291 Complement of A072795.

0, 3, 6, 7, 8, 14, 15, 16, 17, 18, 19, 20, 21, 22, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 0

Antti Karttunen, Mar 17 2003

nonn

This gives positions of those terms in A063171 which begin with digits 11... Also the elements of table A072764 which do not occur in the leftmost column. See the comment at A081292.

Cf. A072795, A014137, A081288.

Cf. A080300, A081289, A081290, A081292.

A014137(n) = sum(k=0, n, binomial(2*k, k)/(k+1););
A081288(n) = my(i=0); while(binomial(2*i, i)/(i+1) <= n, i++); i;
a(n) = if (n==0, 0,  n + A014137(A081288(n)-1)); \\ Michel Marcus, Apr 28 2020

a(0)=0, a(n) = n + A014137(A081288(n)-1).

a(n) = A080300(A081292(n)) = A081289(n) + n - A081290(n).

A081290 a(0) = 0, and for n >=1, a(n) = the largest Catalan number <= n.

Original entry on oeis.org

0, 1, 2, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42

Offset: 0

Antti Karttunen, Mar 17 2003

nonn

After n>0, A000108(n) occurs A000245(n) times.
For all n>0, a(A000108(n)) = A000108(n) [the first occurrence of the n-th Catalan number in this sequence].
Minimal i such that A081289(i) >= A081289(n) [the original definition of the sequence].
In other words, the first position k in A081289 where A081289(n) occurs (the minimal k such that A081289(k) = A081289(n)), and also the first position k in A081288 where A081288(n) occurs (the minimal k such that A081288(k) = A081288(n)). The starting point of the run which contains the n-th term in those sequences.

Cf. A000108, A081288, A081289, A072643, A239903.

Join[{0},With[{catnos=Reverse[CatalanNumber[Range[10]]]},Table[ SelectFirst[ catnos,#<=n&],{n,80}]]] (* This program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, Jul 27 2014 *)

a(0) = 0, a(n) = A000108(A081288(n)-1).

Sum_{n>=1} 1/a(n)^2 = 44*Pi/sqrt(3) - 4*Pi^2 - 38. - Amiram Eldar, Aug 18 2022

Name changed by Antti Karttunen, Apr 26 2014

A081288 a(n) is the minimal i such that A000108(i) > n.

Original entry on oeis.org

0, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Mar 17 2003

nonn

Apart from the initial term 0, each n occurs A000245(n-1) times.

Cf. A000108, A000245, A072643, A081289, A081290. Used to compute A081291.

A081288(n) = my(i=0); while(binomial(2*i, i)/(i+1) <= n, i++); i; \\ Michel Marcus, Apr 28 2020

from sympy import catalan
def a(n):
    if n==0: return 0
    i=1
    while True:
        if catalan(i)>n: return i
        else: i+=1
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 08 2017

A081293 a(n) = A000108(n) + A014137(n).

Original entry on oeis.org

2, 3, 6, 14, 37, 107, 329, 1055, 3486, 11780, 40510, 141286, 498524, 1776312, 6382292, 23097542, 84118037, 308049947, 1133682557, 4190570237, 15551547887, 57919961507, 216418821767, 811055485427, 3047804166425, 11481792822005

Offset: 0

Antti Karttunen, Mar 17 2003

nonn

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Cf. A000108, A014137, A081289.

Cat(n) = binomial(2*n, n)/(n+1); \\ A000108
a(n) = Cat(n)+sum(k=0, n, Cat(k)); \\ Michel Marcus, Apr 28 2020

a(n) = 2*A000108(n) + A014137(n-1). - Michel Marcus, Apr 28 2020

A082855 a(0)=0, a(1)=1, a(n) = A014137(A081288(n-1)-1).

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 9, 9, 9, 9, 9, 9, 9, 9, 9, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

After n>0, A014137(n) occurs A000245(n) times.

Cf. A081289, A082852. A081291(n) = n+A082855(n+1) for n>0.

A014137(n) = sum(k=0, n, binomial(2*k, k)/(k+1););
A081288(n) = my(i=0); while(binomial(2*i, i)/(i+1) <= n, i++); i;
a(n) = if (n<=1, n, A014137(A081288(n-1)-1)); \\ Michel Marcus, Apr 28 2020

(define (A082855 n) (if (< n 2) n (A014137 (-1+ (A081288 (-1+ n))))))

cp's OEIS Frontend

A081291 Complement of A072795.

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A081290 a(0) = 0, and for n >=1, a(n) = the largest Catalan number <= n.

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A081288 a(n) is the minimal i such that A000108(i) > n.

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A081293 a(n) = A000108(n) + A014137(n).

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A082855 a(0)=0, a(1)=1, a(n) = A014137(A081288(n-1)-1).

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