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

0,3

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

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].

Prog

(Scheme, with memoizing definec-macro from Antti Karttunen's IntSeq-library)
;; Version based on direct recurrence:
(definec (A244161 n) (if (zero? n) n (+ (expt 4 (- (A244160 n) 1)) (A244161 (- n (A000108 (A244160 n)))))))
(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

Crossrefs

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

Sequence in context: A347880 A274167 A145265 * A362959 A258935 A275929

Adjacent sequences: A244158 A244159 A244160 * A244162 A244163 A244164

Keyword

nonn

Author

Antti Karttunen, Jun 23 2014

Status

approved