Binomial Coefficient: Formula & Examples
This formula must return an integer for all n , so the radical expression must be an integer otherwise the logarithm does not even return a rational number. No; let's hard-code the 4x4 determinant: We believe that creating an integrated view on the customer is a first step in discovering new fields of opportunities. All the site events are known in advance, but the circle events are not. Thus the Fibonacci sequence is an example of a divisibility sequence.
For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, , , , , …, etc. The ratio of consecutive terms in this sequence shows the same convergence towards the golden ratio. Another consequence is that the limit of the ratio of two Fibonacci numbers offset by a particular finite deviation in index corresponds to the golden ratio raised by that deviation.
Or, in other words:. The resulting recurrence relationships yield Fibonacci numbers as the linear coefficients:. This equation can be proved by induction on n. A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is. From this, the n th element in the Fibonacci series may be read off directly as a closed-form expression:. Equivalently, the same computation may performed by diagonalization of A through use of its eigendecomposition:.
This property can be understood in terms of the continued fraction representation for the golden ratio:. The matrix representation gives the following closed-form expression for the Fibonacci numbers:.
Taking the determinant of both sides of this equation yields Cassini's identity ,. This matches the time for computing the n th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number recursion with memoization.
The question may arise whether a positive integer x is a Fibonacci number. This formula must return an integer for all n , so the radical expression must be an integer otherwise the logarithm does not even return a rational number.
Here, the order of the summand matters. One group contains those sums whose first term is 1 and the other those sums whose first term is 2. It follows that the ordinary generating function of the Fibonacci sequence, i. Numerous other identities can be derived using various methods. Some of the most noteworthy are: The last is an identity for doubling n ; other identities of this type are.
These can be found experimentally using lattice reduction , and are useful in setting up the special number field sieve to factorize a Fibonacci number. The generating function of the Fibonacci sequence is the power series. This can be proved by using the Fibonacci recurrence to expand each coefficient in the infinite sum:. If x is the reciprocal of an integer k that is greater than 1, the closed form of the series becomes. Infinite sums over reciprocal Fibonacci numbers can sometimes be evaluated in terms of theta functions.
For example, we can write the sum of every odd-indexed reciprocal Fibonacci number as. No closed formula for the reciprocal Fibonacci constant.
The Millin series gives the identity . Every third number of the sequence is even and more generally, every k th number of the sequence is a multiple of F k. Thus the Fibonacci sequence is an example of a divisibility sequence. In fact, the Fibonacci sequence satisfies the stronger divisibility property  .
Any three consecutive Fibonacci numbers are pairwise coprime , which means that, for every n ,. These cases can be combined into a single formula, using the Legendre symbol: If n is composite and satisfies the formula, then n is a Fibonacci pseudoprime. When m is large—say a bit number—then we can calculate F m mod n efficiently using the matrix form.
Here the matrix power A m is calculated using modular exponentiation , which can be adapted to matrices. A Fibonacci prime is a Fibonacci number that is prime. The first few are:. Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.
As there are arbitrarily long runs of composite numbers , there are therefore also arbitrarily long runs of composite Fibonacci numbers. The only nontrivial square Fibonacci number is Siksek proved that 8 and are the only such non-trivial perfect powers. Such primes if there are any would be called Wall—Sun—Sun primes.
For odd n , all odd prime divisors of F n are congruent to 1 modulo 4, implying that all odd divisors of F n as the products of odd prime divisors are congruent to 1 modulo 4. Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field.
However, for any particular n , the Pisano period may be found as an instance of cycle detection. Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple.
The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle. The first triangle in this series has sides of length 5, 4, and 3. This series continues indefinitely. The triangle sides a , b , c can be calculated directly:.
The Fibonacci numbers are important in the computational run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: The procedure is illustrated in an example often referred to as the Brock—Mirman economic growth model. Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation , which led to his solving Hilbert's tenth problem.
The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most. Moreover, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.
This is known as Zeckendorf's theorem , and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding.
Fibonacci numbers are used by some pseudorandom number generators. They are also used in planning poker , which is a step in estimating in software development projects that use the Scrum software development methodology. A tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming. Fibonacci numbers arise in the analysis of the Fibonacci heap data structure. The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.
A one-dimensional optimization method, called the Fibonacci search technique , uses Fibonacci numbers. Since the conversion factor 1. To convert from kilometers to miles, shift the register down the Fibonacci sequence instead.
Fibonacci sequences appear in biological settings,  in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem , the fruitlets of a pineapple ,  the flowering of artichoke , an uncurling fern and the arrangement of a pine cone ,  and the family tree of honeybees. A model for the pattern of florets in the head of a sunflower was proposed by H.
The divergence angle, approximately Because this ratio is irrational, no floret has a neighbor at exactly the same angle from the center, so the florets pack efficiently.
Because the rational approximations to the golden ratio are of the form F j: It is often said that sunflowers and similar arrangements have 55 spirals in one direction and 89 in the other or some other pair of adjacent Fibonacci numbers , but this is true only of one range of radii, typically the outermost and thus most conspicuous.
Combinations refer to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k -selection,  k -multiset,  or k -combination with repetition are often used. Although the set of three fruits was small enough to write a complete list of combinations, with large sets this becomes impractical. The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter.
Binomial coefficients can be computed explicitly in various ways. The numerator gives the number of k -permutations of n , i. Finally there is a formula which exhibits this symmetry directly, and has the merit of being easy to remember:. The last formula can be understood directly, by considering the n! Each such permutation gives a k -combination by selecting its first k elements. There are many duplicate selections: From the above formulas follow relations between adjacent numbers in Pascal's triangle in all three directions:.
As a specific example, one can compute the number of five-card hands possible from a standard fifty-two card deck as: Alternatively one may use the formula in terms of factorials and cancel the factors in the numerator against parts of the factors in the denominator, after which only multiplication of the remaining factors is required:.
The reason is that when each division occurs, the intermediate result that is produced is itself a binomial coefficient, so no remainders ever occur. Using the symmetric formula in terms of factorials without performing simplifications gives a rather extensive calculation:. The latter option has the advantage that adding a new largest element to S will not change the initial part of the enumeration, but just add the new k -combinations of the larger set after the previous ones.
Repeating this process, the enumeration can be extended indefinitely with k -combinations of ever larger sets. There are many ways to enumerate k combinations. One way is to visit all the binary numbers less than 2 n. Choose those numbers having k nonzero bits, although this is very inefficient even for small n e. A k - combination with repetitions , or k - multicombination , or multisubset of size k from a set S is given by a sequence of k not necessarily distinct elements of S , where order is not taken into account: In other words, the number of ways to sample k elements from a set of n elements allowing for duplicates i.
The number of multisubsets of size k is then the number of nonnegative integer solutions of the Diophantine equation: This expression, n multichoose k ,  can also be given in terms of binomial coefficients:.