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A356771 a(n) is the sum of the Fibonacci numbers in common in the Zeckendorf and dual Zeckendorf representations of n.

%I A356771 #28 Aug 10 2025 14:12:11
%S A356771 0,1,2,0,4,0,1,7,0,1,2,3,12,0,1,2,0,4,5,6,20,0,1,2,3,4,0,1,7,8,9,10,
%T A356771 11,33,0,1,2,0,4,5,6,7,0,1,2,3,12,13,14,15,13,17,18,19,54,0,1,2,3,4,0,
%U A356771 1,7,8,9,10,11,12,0,1,2,0,4,5,6,20,21,22,23,24
%N A356771 a(n) is the sum of the Fibonacci numbers in common in the Zeckendorf and dual Zeckendorf representations of n.
%C A356771 The Zeckendorf and dual Zeckendorf representations both express a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary (see A022290 for the decoding function):
%C A356771 - in the Zeckendorf representation (or greedy Fibonacci representation):
%C A356771   - Fibonacci numbers are as big as possible (see A035517),
%C A356771   - and the corresponding binary encoding, A003714(n),
%C A356771     cannot have two consecutive 1's;
%C A356771 - in the dual Zeckendorf representation (or lazy Fibonacci representation):
%C A356771   - Fibonacci numbers are as small as possible (see A112309),
%C A356771   - and the corresponding binary encoding, A003754(n+1),
%C A356771     cannot have two consecutive nonleading 0's.
%C A356771 See A356326 for a similar sequence.
%H A356771 Rémy Sigrist, <a href="/A356771/b356771.txt">Table of n, a(n) for n = 0..10946</a>
%H A356771 Rémy Sigrist, <a href="/A356771/a356771_1.gp.txt">PARI program</a>
%H A356771 <a href="/index/Z#Zeckendorf">Index entries for sequences related to Zeckendorf expansion of n</a>
%F A356771 a(n) = A022290(A003714(n) AND A003754(n+1)) (where AND denotes the bitwise AND operator).
%F A356771 a(n) = 0 iff n belongs to A331467.
%F A356771 a(n) = n iff n belongs to A000071.
%e A356771 For n = 28:
%e A356771 - using F(k) = A000045(k),
%e A356771 - the Zeckendorf representation of 28 is F(8) + F(5) + F(3),
%e A356771 - the dual Zeckendorf representation of 28 is F(7) + F(6) + F(5) + F(3),
%e A356771 - F(5) and F(3) appear in both representations,
%e A356771 - so a(28) = F(5) + F(3) = 7.
%o A356771 (PARI) \\ See Links section.
%Y A356771 Cf. A000071, A003714, A003754, A022290, A035517, A112309, A331467, A356326.
%K A356771 nonn,base,look
%O A356771 0,3
%A A356771 _Rémy Sigrist_, Aug 27 2022