In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors omit the initial terms and start the sequence from 1 and 1 or from 1 and 2. Starting from 0 and 1, the next few values in the sequence are:
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation. If the values of the first numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.
François Édouard Anatole Lucas was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him.
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics.
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials.
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes.
In mathematics, the Fibonacci numbers form a sequence defined recursively by:
Willem Abraham Wythoff, born Wijthoff, was a Dutch mathematician.
The Fibonacci Association is a mathematical organization that specializes in the Fibonacci number sequence and a wide variety of related subjects, generalizations, and applications, including recurrence relations, combinatorial identities, binomial coefficients, prime numbers, pseudoprimes, continued fractions, the golden ratio, linear algebra, geometry, real analysis, and complex analysis.
Verner Emil Hoggatt Jr. was an American mathematician, known mostly for his work in Fibonacci numbers and number theory.
Brother Alfred Brousseau, F.S.C., was an educator, photographer and mathematician and was known mostly as a founder of the Fibonacci Association and as an educator.
The Hosoya index, also known as the Z index, of a graph is the total number of matchings in it. The Hosoya index is always at least one, because the empty set of edges is counted as a matching for this purpose. Equivalently, the Hosoya index is the number of non-empty matchings plus one. The index is named after Haruo Hosoya.
In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so that men and women alternate and nobody sits next to his or her partner. This problem was formulated in 1891 by Édouard Lucas and independently, a few years earlier, by Peter Guthrie Tait in connection with knot theory. For a number of couples equal to 3, 4, 5, ... the number of seating arrangements is
In mathematics, a divisibility sequence is an integer sequence indexed by positive integers n such that
In mathematics, the telephone numbers or the involution numbers are a sequence of integers that count the ways n telephone lines can be connected to each other, where each line can be connected to at most one other line. These numbers also describe the number of matchings of a complete graph on n vertices, the number of permutations on n elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with n cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated, giving the values
In mathematics, the Wythoff array is an infinite matrix of integers derived from the Fibonacci sequence and named after Dutch mathematician Willem Abraham Wythoff. Every positive integer occurs exactly once in the array, and every integer sequence defined by the Fibonacci recurrence can be derived by shifting a row of the array.
Joseph Arkin was a mathematician, lecturer and professor at the West Point military academy.
In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers satisfying a linear recurrence relation: each number in the sequence is equal to a fixed linear combination of one or more of its immediate predecessors. A constant-recursive sequence is also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, or C-finite sequence.
In mathematics, the fibbinary numbers are the numbers whose binary representation does not contain two consecutive ones. That is, they are sums of distinct and non-consecutive powers of two.
