A greedy algorithm for finding a representation of the number of garafibonacci numbers?

Question

According to the Zeckendorf theorem , each positive integer can be written in a unique way as the sum of inconsistent different Fibonacci numbers. Such a decomposition can be easily found using the greedy algorithm, which consists essentially in subtracting the largest Fibonacci number, which is suitable and iterated, for example:

20 = 13 + 7 = 13 + 5 + 2

Nevertheless, it also follows from the theorem that any integer (also & lt = 0) has a unique expansion in the form of the sum of various non-negative numbers of negative Fibonacci numbers, i.e. a sequence

0, 1, -1, 2, -3, 5, -8, ...

or F_ (- n) = (-1) ^ (n + 1) F_n. Some examples:

-4 = - 3 - 1

4 = 5 + 1

11 = 13 - 3 + 1

Is there a well-known simple algorithm for decomposing a given integer in this way?

Answer 1

, negafibonacci.

, , ( ) ( ). ,

• F ₀ + 1 F ₂, ,
• F ₂ + 1 F ₄, ,
• F ₄ + 1 F ₆, ,
• F ₆ + 1 F ₈, ,
• F ₈ + 1 F ₁₀, ,
• ...
• F _2k + 1 to F _{2k + 2}, ,
• ...

n , :

• -F ₁ -F ₃ + 1, ,
• -F ₃ -F ₅ + 1, ,
• -F ₅ -F ₇ + 1, ,
• -F ₇ -F ₉ + 1, ,
• ...
• -F _2k-1 -F _{2k + 1} + 1, ,
• ...

:

• , [F _2k + 1, F _{2k + 2}], n, F _{2k + 1} F _{2k + 1} n.
• , [-F _2k-1, -F _{2k + 1} + 1], n, add -F _2k, F _2k .
• , .

. , 27 negafibonacci. , 21 + 1 & le; 27 & le; 55. ( ) 34, 34 27-34 = -7 negafibonacci.

, , 5 + 1 & le; 7 & le; 13, 7 , ( ) 8. -8 1 negafibonacci.

, 0 + 1 & le; 1 & le; 1, 1 , (-) 1. 1 0 negafibonacci.

0, ! ! 34 - 8 + 1 = 27.

. -, , , ( - F _{2k + 1}), , ( - -F _2k). .

. . , [F _2k + 1, F _{2k + 2}], F _{2k + 1 > . F 2k + 2 - F 2k + 1= F 2k, , , [F 2k-2 + 1, F 2k], , , F 2k-1. , , , .}

F _2k + 1 - F _{2k + 1}= -F _2k-1 + 1. , , " " , , [F _2k-1 + 1, F _2k-3], , , - F _2k-2. .

, , .

, (F _k-1, F _k, F _{k + 1}) ( ), , n. (, ) 0, . , O (log n).