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A244161
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Greedy Catalan Base (A014418) interpreted as base-4 numbers, then shown in decimal.
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10
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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
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OFFSET
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0,3
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COMMENTS
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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".
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LINKS
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FORMULA
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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].
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PROG
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(Scheme, with memoizing definec-macro from Antti Karttunen's IntSeq-library)
;; Version based on direct recurrence:
(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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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