A244161 Greedy Catalan Base (A014418) interpreted as base-4 numbers, then shown in decimal.

0, 1, 4, 5, 8, 16, 17, 20, 21, 24, 32, 33, 36, 37, 64, 65, 68, 69, 72, 80, 81, 84, 85, 88, 96, 97, 100, 101, 128, 129, 132, 133, 136, 144, 145, 148, 149, 152, 160, 161, 164, 165, 256, 257, 260, 261, 264, 272, 273, 276, 277, 280, 288, 289, 292, 293, 320, 321, 324, 325

Offset: 0

Author

Antti Karttunen, Jun 23 2014

Keywords

• nonn

Comments

This representation does not lose any information, because C(n+1)/C(n) [where C(n) is the n-th Catalan number, A000108(n)] approaches 4 from below, but never attains it.

Analogously to "Fibbinary numbers", A003714, this sequence could be called "Catquaternary numbers".

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A014418, A003714, A085183, A085184, A244160.

Programs

Python

from sympy import catalan
def a244160(n):
    if n==0: return 0
    i=1
    while True:
        if catalan(i)>n: break
        else: i+=1
    return i - 1
def a(n):
    if n==0: return 0
    x=a244160(n)
    return 4**(x - 1) + a(n - catalan(x))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 08 2017

Formula

a(0) = 0, a(n) = 4^(A244160(n)-1) + a(n-A000108(A244160(n))). [Where A244160 gives the index of the largest Catalan number that still fits into the sum].

A000035(a(n)) = A000035(A014418(n)). [This sequence and the base-10 version are equal when reduced modulo 2].