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In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the n-th is equal to the (n + 2)-th Fibonacci number minus 1. [33] In symbols: = = + This may be seen by dividing all sequences summing to + based on the location of the first 2.
where F n is the n th Fibonacci number. Such a sum is called the Zeckendorf representation of N. The Fibonacci coding of N can be derived from its Zeckendorf representation. For example, the Zeckendorf representation of 64 is 64 = 55 + 8 + 1. There are other ways of representing 64 as the sum of Fibonacci numbers 64 = 55 + 5 + 3 + 1 64 = 34 ...
A Fibonacci sequence of order n is an integer sequence in which each sequence element is the sum of the previous elements (with the exception of the first elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2.
The reciprocal Fibonacci constant is the sum of the reciprocals of the Fibonacci numbers, which is known to be finite and irrational and approximately equal to 3.3599 . For other finite sums of subsets of the reciprocals of Fibonacci numbers, see here.
The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers: = = = + + + + + + + +. Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.
Each positive integer n has 2 n−1 distinct compositions. Bijection between 3 bit binary numbers and compositions of 4. A weak composition of an integer n is similar to a composition of n, but allowing terms of the sequence to be zero: it is a way of writing n as the sum of a sequence of non-negative integers. As a consequence every positive ...
A prime divides if and only if p is congruent to ±1 modulo 5, and p divides + if and only if it is congruent to ±2 modulo 5. (For p = 5, F 5 = 5 so 5 divides F 5) . Fibonacci numbers that have a prime index p do not share any common divisors greater than 1 with the preceding Fibonacci numbers, due to the identity: [6]
Repeat the previous steps, substituting the remainder for N, until a remainder of 0 is reached. Place an additional 1 after the rightmost digit in the code word. To decode a code word, remove the final "1", assign the remaining the values 1,2,3,5,8,13... (the Fibonacci numbers) to the bits in the code word, and sum the values of the "1" bits.