1.
Recursion
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Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic, the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines a number of instances, it is often done in such a way that no loop or infinite chain of references can occur. The ancestors of ones ancestors are also ones ancestors, the Fibonacci sequence is a classic example of recursion, Fib =0 as base case 1, Fib =1 as base case 2, For all integers n >1, Fib, = Fib + Fib. Many mathematical axioms are based upon recursive rules, for example, the formal definition of the natural numbers by the Peano axioms can be described as,0 is a natural number, and each natural number has a successor, which is also a natural number. By this base case and recursive rule, one can generate the set of all natural numbers, recursively defined mathematical objects include functions, sets, and especially fractals. There are various more tongue-in-cheek definitions of recursion, see recursive humor, Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be recursive, to understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, the running of a procedure involves actually following the rules and performing the steps. An analogy, a procedure is like a recipe, running a procedure is like actually preparing the meal. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure. For instance, a recipe might refer to cooking vegetables, which is another procedure that in turn requires heating water, for this reason recursive definitions are very rare in everyday situations. An example could be the procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point, If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively, if every trial fails by reaching only dead ends, return on the path led to this branching point. Whether this actually defines a terminating procedure depends on the nature of the maze, in any case, executing the procedure requires carefully recording all currently explored branching points, and which of their branches have already been exhaustively tried. This can be understood in terms of a definition of a syntactic category. A sentence can have a structure in which what follows the verb is another sentence, Dorothy thinks witches are dangerous, so a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, and optionally another sentence

2.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra

3.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently

4.
Padovan sequence
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The Padovan sequence is the sequence of integers P defined by the initial values P = P = P =1, and the recurrence relation P = P + P. The first few values of P are 1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265. The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom, Hans van der Laan, Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996 and he also writes about it in one of his books, Math Hysteria, Fun Games With Mathematics. The above definition is the one given by Ian Stewart and by MathWorld, other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets. This is a property of recurrence relations, the Perrin sequence can be obtained from the Padovan sequence by the following formula, P e r r i n = P + P. e. The Padovan sequence also satisfies the identity P2 − P P = P. The Padovan sequence is related to sums of binomial coefficients by the following identity, P = ∑2 m + n = k = ∑ m = ⌈ k /3 ⌉ ⌊ k /2 ⌋. For example, for k =12, the values for the pair with 2m + n =12 which give non-zero binomial coefficients are, and, and, + + =1 +10 +1 =12 = P. The Padovan sequence numbers can be written in terms of powers of the roots of the equation x 3 − x −1 =0 and this equation has 3 roots, one real root p and two complex conjugate roots q and r. Given these three roots, the Padovan sequence can be expressed by a formula involving p, q and r, P = a p n + b q n + c r n where a, b and c are constants. Since the magnitudes of the complex roots q and r are both less than 1, the powers of these roots approach 0 for large n, and P − a p n tends to zero. For all n ≥0, P is the integer closest to p n −1 s, the ratio of successive terms in the Padovan sequence approaches p, which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the ratio does to the Fibonacci sequence. P is the number of ways of writing n +2 as a sum in which each term is either 2 or 3. This can be used to prove identities involving products of the Padovan sequence with geometric terms, such as, ∑ n =0 ∞ P α n = α2 α3 − α −1. A Padovan prime is P that is prime, the first few Padovan primes are 2,3,5,7,37,151,3329,23833. Also, if you count the number of As, Bs and Cs in each string, then for the nth string, you have P As, P Bs, the count of BB pairs, AA pairs and CC pairs are also Padovan numbers

5.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory

6.
Stella octangula number
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In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form n. The sequence of stella octangula numbers is 0,1,14,51,124,245,426,679,1016,1449,1990. There are only two positive square stella octangula numbers,1 and 9653449 =31072 =2, corresponding to n =1 and n =169 respectively. The elliptic curve describing the square stella octangula numbers, m 2 = n may be placed in the equivalent Weierstrass form x 2 = y 3 −2 y by the change of variables x = 2m, y = 2n. Louis J. Mordell conjectured that the proof could be simplified, weisstein, Eric W. Stella Octangula Number

7.
Exponentiation
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Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5

8.
Square number
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In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number

9.
Cube (algebra)
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In arithmetic and algebra, the cube of a number n is its third power, the result of the number multiplied by itself twice, n3 = n × n × n. It is also the number multiplied by its square, n3 = n × n2 and this is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called extracting the cube root of n and it determines the side of the cube of a given volume. It is also n raised to the one-third power, both cube and cube root are odd functions,3 = −. The cube of a number or any other mathematical expression is denoted by a superscript 3, a cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer. The perfect cubes up to 603 are, Geometrically speaking, an integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger. For example,27 small cubes can be arranged into one larger one with the appearance of a Rubiks Cube, the difference between the cubes of consecutive integers can be expressed as follows, n3 −3 = 3n +1. There is no minimum perfect cube, since the cube of an integer is negative. For example, −4 × −4 × −4 = −64, unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25,75 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6, some cube numbers are also square numbers, for example,64 is a square number and a cube number. This happens if and only if the number is a perfect sixth power, the last digits of each 3rd power are, It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1,8 or 9. That is their values modulo 9 may be only −1,1 and 0, every positive integer can be written as the sum of nine positive cubes. The equation x3 + y3 = z3 has no solutions in integers. In fact, it has none in Eisenstein integers, both of these statements are also true for the equation x3 + y3 = 3z3. The sum of the first n cubes is the nth triangle number squared,13 +23 + ⋯ + n 3 =2 =2. Proofs Charles Wheatstone gives a simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. Indeed, he begins by giving the identity n 3 = + + + ⋯ + ⏟ n consecutive odd numbers, kanim provides a purely visual proof, Benjamin & Orrison provide two additional proofs, and Nelsen gives seven geometric proofs

10.
Pell number
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In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1,2,5,12, and 29. The numerators of the sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers, these numbers form a second infinite sequence that begins with 2,6,14,34. As with Pells equation, the name of the Pell numbers stems from Leonhard Eulers mistaken attribution of the equation, the Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type, the Pell and companion Pell numbers are Lucas sequences. The Pell numbers are defined by the recurrence relation P n = {0 if n =0,1 if n =1,2 P n −1 + P n −2 otherwise. In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are 0,1,2,5,12,29,70,169,408,985,2378,5741,13860, …. The Pell numbers can also be expressed by the closed form formula P n = n − n 22, a third definition is possible, from the matrix formula = n. Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation x 2 −2 y 2 = ±1 and that is, the solutions have the form P n −1 + P n P n. The approximation 2 ≈577408 of this type was known to Indian mathematicians in the third or fourth century B. C, the Greek mathematicians of the fifth century B. C. also knew of this sequence of approximations, Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and these approximations can be derived from the continued fraction expansion of 2,2 =1 +12 +12 +12 +12 +12 + ⋱. As Knuth describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to an octagon with vertex coordinates. All vertices are equally distant from the origin, and form uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are equally distant from the origin. A Pell prime is a Pell number that is prime, the first few Pell primes are 2,5,29,5741, …. The indices of these primes within the sequence of all Pell numbers are 2,3,5,11,13,29,41,53,59,89,97,101,167,181,191, … These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, the only Pell numbers that are squares, cubes, or any higher power of an integer are 0,1, and 169 =132. However, despite having so few squares or other powers, Pell numbers have a connection to square triangular numbers

11.
Achilles number
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An Achilles number is a number that is powerful but not a perfect power. A positive integer n is a number if, for every prime factor p of n. In other words, every prime factor appears at least squared in the factorization, however, not all powerful numbers are Achilles numbers, only those that cannot be represented as mk, where m and k are positive integers greater than 1. Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, strong Achilles numbers are Achilles numbers whose Euler totients are also Achilles numbers. A number n = p1a1 p2a2 … pkak is powerful if min ≥2, if in addition gcd =1 the number is an Achilles number. The smallest pair of consecutive Achilles numbers is,5425069447 =73 ×412 ×9725425069448 =23 ×260412108 is a powerful number and its prime factorization is 22 ·33, and thus its prime factors are 2 and 3. Both 22 =4 and 32 =9 are divisors of 108, however,108 cannot be represented as mk, where m and k are positive integers greater than 1, so 108 is an Achilles number. 360 is not an Achilles number because it is not powerful, one of its prime factors is 5 but 360 is not divisible by 52 =25. Finally,784 is not an Achilles number and it is a powerful number, because not only are 2 and 7 its only prime factors, but also 22 =4 and 72 =49 are divisors of it. Nonetheless, it is a power,784 =24 ⋅72 =2 ⋅72 =2 =282. So it is not an Achilles number