Fibonacci number in the context of "Liber Abaci"

Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of experiments to test scientific theories. The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side. Theoretical biology focuses more on the development of theoretical principles for biology while mathematical biology focuses on the use of mathematical tools to study biological systems, even though the two terms interchange; overlapping as Artificial Immune Systems of Amorphous Computation.

Mathematical biology aims at the mathematical representation and modeling of biological processes, using techniques and tools of applied mathematics. It can be useful in both theoretical and practical research. Describing systems in a quantitative manner means their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter; requiring mathematical models.

In mathematics, a recurrence relation is an equation according to which the ${\displaystyle n}$th term of a sequence of numbers is equal to some combination of the previous terms. Often, only ${\displaystyle k}$ previous terms of the sequence appear in the equation, for a parameter ${\displaystyle k}$ that is independent of ${\displaystyle n}$; this number ${\displaystyle k}$ is called the order of the relation. If the values of the first ${\displaystyle k}$ numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.

In linear recurrences, the nth term is equated to a linear function of the ${\displaystyle k}$ previous terms. A famous example is the recurrence for the Fibonacci numbers,$${\displaystyle F_{n}=F_{n-1}+F_{n-2}}$$where the order ${\displaystyle k}$ is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on ${\displaystyle n.}$ For these recurrences, one can express the general term of the sequence as a closed-form expression of ${\displaystyle n}$. As well, linear recurrences with polynomial coefficients depending on ${\displaystyle n}$ are also important, because many common elementary functions and special functions have a Taylor series whose coefficients satisfy such a recurrence relation (see holonomic function).

In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff). Some examples of recursively definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set.

A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the factorial function n! is defined by the rules

Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models, and abstractions of living organisms to investigate the principles that govern the structure, development, and behavior of the systems. In contrast, experimental biology involves the conduction of experiments to test scientific theories. The field is sometimes called mathematical biology or biomathematics to emphasize the mathematical aspect, or as theoretical biology to highlight the biological aspect. Theoretical biology focuses more on the development of theoretical principles for biology, while mathematical biology focuses on the application of mathematical tools to study biological systems. However, these terms are often used interchangeably, merging into the concept of Artificial Immune Systems of Amorphous Computation.

Mathematical biology aims at the mathematical representation and modeling of biological processes, using techniques and tools of applied mathematics. It can be useful in both theoretical and practical research. Describing systems in a quantitative manner means their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter; requiring mathematical models.