Pell numbers are a sequence of integers defined by a specific recurrence relation. The Pell numbers are similar to the Fibonacci numbers but are defined differently. The sequence starts with initial values, and each subsequent number is derived from the previous two.

In number theory and combinatorics, the **partition function** is a function that counts the number of distinct ways a given positive integer can be expressed as a sum of positive integers, regardless of the order of addends.

The Padovan sequence is a sequence of numbers defined by a specific recurrence relation.

The Ordered Bell number is a concept in combinatorial mathematics that counts the number of ways to partition a set into a certain number of non-empty ordered subsets. More formally, the \( n \)-th Ordered Bell number, denoted as \( B_n^{o} \), gives the number of ways to partition a set of size \( n \) into \( k \) non-empty subsets, where the order of the subsets matters.

The On-Line Encyclopedia of Integer Sequences (OEIS) is a comprehensive database that collects and catalogs integer sequences. Launched in 1964 by Neil J. A. Sloane, the OEIS has grown significantly over the years and is now a valuable resource for mathematicians, scientists, and hobbyists interested in number theory, combinatorics, and other areas involving sequences of integers.

An odious number is a non-negative integer that has an odd number of 1s in its binary representation. In contrast, a number that has an even number of 1s in its binary form is referred to as an "elegant number." For example: - The number 3 in binary is `11`, which contains two 1s (an even number), so it is not odious.

A **nontotient** is a positive integer \( n \) for which there is no integer \( k \) such that \( k \) and \( n \) are coprime, and \( \phi(k) = n \), where \( \phi \) is the Euler's totient function. The Euler's totient function \( \phi(k) \) counts the number of integers up to \( k \) that are coprime to \( k \).

A nonhypotenuse number is not a standard term in mathematics, so its meaning may vary depending on context. However, it could be inferred as a number that cannot be the length of the hypotenuse of a right triangle, based on the properties of right triangles in Euclidean geometry.

The noncototient is a mathematical concept related to number theory. Specifically, it refers to the integers \( n \) for which the equation \( \phi(m) = n \) has no solution for any integer \( m \). Here, \( \phi(m) \) is the Euler's totient function, which counts the number of positive integers up to \( m \) that are relatively prime to \( m \).

The Narayana numbers are a sequence of numbers that appear in combinatorial mathematics and are related to various counting problems, including those involving paths and combinations.

The Ménage problem is a classic problem in combinatorics that involves counting the number of ways to arrange couples such that no couple sits next to each other. Typically, the problem is stated with a specific number of couples, and the arrangements are considered around a circular table.

A multiplicative partition of a positive integer is a way to express that integer as a product of positive integers, where the order of the factors does not matter. In other words, it refers to breaking down a number into factors such that their product equals the original number.

The Motzkin numbers are a sequence of natural numbers that arise in various combinatorial contexts. The \(n\)th Motzkin number, denoted as \(M_n\), counts the number of ways to draw non-intersecting chords connecting \(n\) points on a circle to the diameter below, without any chords crossing each other. Additionally, it can represent the number of monotonic paths along the edges of a grid.

The Mian–Chowla sequence is an infinite sequence of integers defined by a specific recursive relationship. The sequence is constructed in such a way that it avoids repetitions and maintains specific properties regarding sums of elements. The definition of the Mian–Chowla sequence can be outlined as follows: 1. The first element of the sequence is 1, i.e., \( a_1 = 1 \).

A Mersenne prime is a specific type of prime number that can be expressed in the form \(M_n = 2^n - 1\), where \(n\) is a positive integer. In other words, if \(M_n\) is prime, then \(n\) itself must also be prime.

In mathematics, a "meander" refers to a specific type of curve or path that has a winding, zigzagging shape. More formally, a meander can be described in the context of topology and combinatorial geometry, where it often pertains to the study of curves on a plane that cross themselves in a certain way. A classic example of meanders arises in the study of river paths or the trajectory of flowing water, which tend to form intricate, looping patterns as they navigate through landscapes.

In physics, the term "magic number" refers to specific numbers of nucleons (protons and neutrons) in atomic nuclei that result in a nucleus being more stable than others. These magic numbers correspond to closed shells of nucleons, similar to how noble gases have filled electron shells, leading to their stability.

A **magic constant** is the sum of the numbers in any row, column, or diagonal of a magic square. A magic square is a grid arrangement of distinct integers such that the sum of the numbers in each row, column, and both main diagonals is the same.

A Löschian number refers to a specific type of number in number theory that is connected to the properties of the Löschian polynomial. The term itself may not be widely recognized, as Löschian numbers are not a standard concept in mathematics like prime numbers or Fibonacci numbers.

Lucky numbers are a sequence of natural numbers that are generated by a specific sieve process, first introduced by the mathematician Leonhard Euler. The process of generating lucky numbers is similar to that used in the Sieve of Eratosthenes for finding prime numbers, but instead of eliminating multiples of prime numbers, it eliminates numbers based on their positions.