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

Author

Antti Karttunen, Jun 23 2014

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

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

Crossrefs

After zero, one less than A081288.

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

Programs

Mathematica

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 *)

Python

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

Scheme

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

Formula

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

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