A244215 -id:A244215 - cp's OEIS frontend

A244160 a(0)=0, and for n >= 1, a(n) = the largest k such that k-th Catalan number <= n.

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 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, 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, 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, 5, 5, 5, 5, 5, 5, 6

Offset: 0

Antti Karttunen, Jun 23 2014

nonn

Apart from 0, each n occurs A000245(n) times.

For n >= 1, a(n) gives the largest k such that C(k) <= n, where C(k) stands for the k-th Catalan number, A000108(k).

			For n=1, the largest k such that C(k) <= 1 is 1, thus a(1) = 1.
For n=2, the largest k such that C(k) <= 2 is 2, thus a(2) = 2.
For n=3, the largest k such that C(k) <= 3 is 2, thus a(3) = 2.
For n=4, the largest k such that C(k) <= 4 is 2, thus a(4) = 2.
For n=5, the largest k such that C(k) <= 5 is 3, thus a(5) = 3.

Michael De Vlieger, Table of n, a(n) for n = 0..4861

After zero, one less than A081288.

Cf. A000108, A000245, A081290, A014418, A239903, A244215, A244159, A236859, A126307.

MapIndexed[ConstantArray[First@ #2 - 1, #1] &, Differences@ Array[CatalanNumber, 8, 0]] /. {} -> {0} // Flatten (* Michael De Vlieger, Jun 08 2017 *)
Join[{0},Table[PadRight[{},CatalanNumber[n+1]-CatalanNumber[n],n],{n,6}]// Flatten] (* Harvey P. Dale, Aug 23 2021 *)

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

(define (A244160 n) (if (zero? n) n (- (A081288 n) 1)))

a(0) = 0, and for n>=1, a(n) = A081288(n)-1.

For all n>=1, A000108(a(n)) = A081290(n).

A244217 Numbers n less than twice the largest Catalan number that is less than or equal to n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 132, 133, 134

Offset: 1

Antti Karttunen, Jun 23 2014

nonn

Equally: Numbers k such that if m is the largest Catalan number <= k [= A081290(k)], then k < 2*m.

Numbers k such that the Greedy Catalan Base representation of k (A014418(k)) starts with digit 1.

			41 is not a member, because the largest Catalan number less than or equal to 41 is C(4) = 14, and 41 is not less than 2*14 = 28.
42 is a member, because the largest Catalan number less than or equal to 42 is C(5) = 42 itself, and 42 certainly is less than 2*42 = 84.
See also the examples given for the complement of this sequence: A244216.

Antti Karttunen, Table of n, a(n) for n = 1..6918

Positions of nonzeros in A244215.
Complement of A244216.
Cf. A000108 (a subsequence), A014418, A081290, A244314.

A244216 Numbers n that are at least twice the size the largest Catalan number less than or equal to n.

Original entry on oeis.org

4, 10, 11, 12, 13, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 264

Offset: 1

Antti Karttunen, Jun 23 2014

nonn

Equally: Numbers k such that if m is the largest Catalan number <= k [= A081290(k)], then k >= 2*m.

Numbers k such that the Greedy Catalan Base representation of k (cf. A014418) starts with digit 2 or 3.

			4 is present, as the largest Catalan number (A000108(k), here C(k)) less than or equal to 4 is C(2) = 2, and 4 >= 2*2.
5 is not present, as the largest Catalan number <= 5 is C(3) = 5, and 5 < 2*5.
10 is present, as the largest Catalan number <= 10 is C(3) = 5, and 10 >= 2*5.
11 is present, as the largest Catalan number <= 11 is C(3) = 5, and 11 >= 2*5.
27 is not present, as the largest Catalan number <= 27 is C(4) = 14, and 27 < 2*14.
28 is present, as the largest Catalan number <= 28 is C(4) = 14, and 28 >= 2*14.

Antti Karttunen, Table of n, a(n) for n = 1..9878

Positions of zeros in A244215. Complement of A244217.

Cf. A000108, A014418, A081290.

cp's OEIS Frontend

A244160 a(0)=0, and for n >= 1, a(n) = the largest k such that k-th Catalan number <= n.

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A244217 Numbers n less than twice the largest Catalan number that is less than or equal to n.

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A244216 Numbers n that are at least twice the size the largest Catalan number less than or equal to n.

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Author

Keywords

Comments

Examples

Links

Crossrefs