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Greedy Catalan Base (A014418) interpreted as base-4 numbers, then shown in decimal.
10

%I #19 Mar 24 2021 09:52:21

%S 0,1,4,5,8,16,17,20,21,24,32,33,36,37,64,65,68,69,72,80,81,84,85,88,

%T 96,97,100,101,128,129,132,133,136,144,145,148,149,152,160,161,164,

%U 165,256,257,260,261,264,272,273,276,277,280,288,289,292,293,320,321,324,325

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

%C 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.

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

%H Indranil Ghosh, <a href="/A244161/b244161.txt">Table of n, a(n) for n = 0..1000</a>

%F 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].

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

%o (Scheme, with memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)

%o ;; Version based on direct recurrence:

%o (definec (A244161 n) (if (zero? n) n (+ (expt 4 (- (A244160 n) 1)) (A244161 (- n (A000108 (A244160 n)))))))

%o (Python)

%o from sympy import catalan

%o def a244160(n):

%o if n==0: return 0

%o i=1

%o while True:

%o if catalan(i)>n: break

%o else: i+=1

%o return i - 1

%o def a(n):

%o if n==0: return 0

%o x=a244160(n)

%o return 4**(x - 1) + a(n - catalan(x))

%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 08 2017

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

%K nonn

%O 0,3

%A _Antti Karttunen_, Jun 23 2014