A356771 - OEIS

%I #22 Sep 06 2022 10:29:15

%S 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 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 1,7,8,9,10,11,12,0,1,2,0,4,5,6,20,21,22,23,24

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

%C 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 - in the Zeckendorf representation (or greedy Fibonacci representation):

%C   - Fibonacci numbers are as big as possible (see A035517),

%C   - and the corresponding binary encoding, A003714(n),

%C     cannot have two consecutive 1's;

%C - in the dual Zeckendorf representation (or lazy Fibonacci representation):

%C   - Fibonacci numbers are as small as possible (see A112309),

%C   - and the corresponding binary encoding, A003754(n+1),

%C     cannot have two consecutive nonleading 0's.

%C See A356326 for a similar sequence.

%H Rémy Sigrist, <a href="/A356771/b356771.txt">Table of n, a(n) for n = 0..10946</a>

%H Rémy Sigrist, <a href="/A356771/a356771_1.gp.txt">PARI program</a>

%H <a href="/index/Z#Zeckendorf">Index entries for sequences related to Zeckendorf expansion of n</a>

%F a(n) = A022290(A003714(n) AND A003754(n+1)) (where AND denotes the bitwise AND operator).

%F a(n) = 0 iff n belongs to A331467.

%F a(n) = n iff n belongs to A000071.

%e For n = 28:

%e - using F(k) = A000045(k),

%e - the Zeckendorf representation of 28 is F(8) + F(5) + F(3),

%e - the dual Zeckendorf representation of 28 is F(7) + F(6) + F(5) + F(3),

%e - F(5) and F(3) appear in both representations,

%e - so a(28) = F(5) + F(3) = 7.

%o (PARI) See Links section.

%Y Cf. A000071, A003714, A003754, A022290, A035517, A112309, A331467, A356326.

%K nonn,base

%O 0,3

%A _Rémy Sigrist_, Aug 27 2022