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The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation(1)Since the diamond is a rhombus with diagonals and , it has inradius(2)(3)Writing as an algebraic curve gives the quartic curve(4)which is a diamond curve with the diamond edges extended to infinity.When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).The diamond is also the name given to the unique 2-polyiamond...

The problem of determining how many nonattacking kings can be placed on an chessboard. For , the solution is 16, as illustrated above (Madachy 1979). In general, the solutions are(1)(Madachy 1979), giving the sequence of doubled squares 1, 1, 4, 4, 9, 9, 16, 16, ... (OEIS A008794). This sequence has generating function(2)The minimal number of kings needed to occupy or attack every square on an chessboard (i.e., domination numbers for the king graphs) are given for , 2, ... by 1, 1, 1, 4, 4, 4, 9, 9, 9, 16, ... (OEIS A075561), with the case illustrated above and noted by (Madachy 1979, p. 39). In general, for an chessboard,(3)

If the three straight lines joining the corresponding vertices of two triangles and all meet in a point (the perspector), then the three intersections of pairs of corresponding sides lie on a straight line (the perspectrix). Equivalently, if two triangles are perspective from a point, they are perspective from a line.The 10 lines and 10 3-line intersections form a configuration sometimes called Desargues' configuration.Desargues' theorem is self-dual.

Consider an (0, 1)-matrix such as(1)for . Call two elements adjacent if they lie in positions and , and , or and for some . Call the number of such arrays with no pairs of adjacent 1s. Equivalently, is the number of configurations of nonattacking kings on an chessboard with regular hexagonal cells.The first few values of for , 2, ... are 2, 6, 43, 557, 14432, ... (OEIS A066863).The hard square hexagon constant is then given by(2)(3)(OEIS A085851).Amazingly, is algebraic and is given by(4)where(5)(6)(7)(8)(9)(10)(11)(Baxter 1980, Joyce 1988ab).The variable can be expressed in terms of the tribonacci constant(12)where is a polynomial root, as(13)(14)(15)(T. Piezas III, pers. comm., Feb. 11, 2006).Explicitly, is the unique positive root(16)where denotes the th root of the polynomial in the ordering of the Wolfram Language...

A projective plane, sometimes called a twisted sphere (Henle 1994, p. 110), is a surface without boundary derived from a usual plane by addition of a line at infinity. Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet. A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist. It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself.A finite projective plane of order is formally defined as a set of points with the properties that: 1. Any two points determine a line,2. Any two lines determine a point,3. Every point has lines on it, and 4. Every line contains points. (Note that some of these properties are redundant.) A projective plane is therefore a symmetric (, , 1)..

A nexus number is a figurate number built up of the nexus of cells less than steps away from a given cell. The th -dimensional nexus number is given by(1)(2)where is a binomial coefficient. The symbolic representations and sequences for first few -dimensional nexus numbers are given in the table below.name01unit1odd number2hex number3rhombic dodecahedral number4nexus numberOEIS, , , ...01, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...1A0054081, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...2A0032151, 7, 19, 37, 61, 91, 127, 169, 217, ...3A0059171, 15, 65, 175, 369, 671, 1105, 1695, 2465, ...4A0225211, 31, 211, 781, 2101, 4651, 9031, 15961, ...

The -roll mill curve is given by the equationwhere is a binomial coefficient.

The th central fibonomial coefficient is defined as(1)(2)where is a fibonomial coefficient, is a Fibonacci number, is the golden ratio, and is a q-Pochhammer symbol (E. W. Weisstein, Dec. 8, 2009).For , 2, ..., the first few are 1, 6, 60, 1820, 136136, ... (OEIS A003267).

As originally stated by Gould (1972),(1)where GCD is the greatest common divisor and is a binomial coefficient. This was subsequently extended by D. Singmaster to(2)(Sato 1975), and generalized by Sato (1975) to(3)An even larger generalization was obtained by Hitotumatu and Sato (1975), who defined(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)with(16)and showed that each of the twelve binomial coefficients , , , , , , , , , , , and has equal greatest common divisor.A second star of David theorem states that if two triangles are drawn centered on a given element of Pascal's triangle as illustrated above, then the products of the three numbers in the associated points of each of the two stars are the same (Butterworth 2002). This follows from the fact that(17)(18)(19)The second star of David theorem holds true not only for the usual binomial coefficients, but also for q-binomial coefficients, where the common product is given by(20)In..

The idempotent numbers are given bywhere is a Bell polynomial and is a binomial coefficient. A table of the first few is given below.A000027A001788A036216A040075A050982A050988A050989112213361442412155809020166240540240301776722835224052542188179213608179207000100856994608612361290247875018144176410101152026244086016078750027216041160

(1)The number triangle illustrated above (OEIS A008949) composed of the partial sums of binomial coefficients,(2)(3)where is a gamma function and is a hypergeometric function.The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Bernoulli triangle.

The fibonomial coefficient (sometimes also called simply the Fibonacci coefficient) is defined by(1)where and is a Fibonacci number. This coefficient satisfies(2)for , where is a Lucas number.The triangle of fibonomial coefficients is given by(3)(OEIS A010048). may be called the central fibonomial coefficient by analogy with the central binomial coefficient.

Pascal's triangle is a number triangle with numbersarranged in staggered rows such that(1)where is a binomial coefficient. The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám. It is therefore known as the Yanghui triangle in China. Starting with , the triangle is(2)(OEIS A007318). Pascal's formula shows that each subsequent row is obtained by adding the two entries diagonally above,(3)The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Pascal's triangle.The first number after the 1 in each row divides all other numbers in that row iff it is a prime.The sums of the number of odd entries in the first rows of Pascal's triangle for , 1, ... are 0, 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, 37, 45, 49, ... (OEIS A006046). It is then..

The nesting of two or more functions to form a single new function is known as composition. The composition of two functions and is denoted , where is a function whose domain includes the range of . The notation(1)is sometimes used to explicitly indicate the variable.Composition is associative, so that(2)If the functions is continuous at and is continuous at , then is also continuous at .A function which is the composition of two other functions, say and , is sometimes said to be a composite function.Faà di Bruno's formula gives an explicit formula for the th derivative of the composition .A combinatorial composition is defined as an ordered arrangement of nonnegative integers which sum to (Skiena 1990, p. 60). It is therefore a partition in which order is significant. For example, there are eight compositions of 4,(3)(4)(5)(6)(7)(8)(9)(10)A positive integer has compositions.The number of compositions of into parts (where..

For a permutation in the symmetric group , the -permutation graph of a labeled graph is the graph union of two disjoint copies of (say, and ), together with the lines joining point of with of (Harary 1994, p. 175).Skiena (1990, p. 28) defines a permutation graph as a graph whose edges correspond exactly to being a permutation inversion is some permutation , i.e., but occurs before in . The above graph corresponds to the permutation , which has permutation inversion . Permutation graphs are implemented as PermutationGraph[p] in the Wolfram Language package Combinatorica` .

A transposition graph is a graph whose nodes correspond to permutations and edges to permutations that differ by exactly one transposition (Skiena 1990, p. 9, Clark 2005).The transposition graph has vertex count , edge count (for ), and is regular of degree (Clark 2005). All cycles in transposition graphs are of even length, making them bipartite.The transposition graph of a multiset is always Hamiltonian(Chase 1973).Special cases are summarized in the table below.1singleton graph 22-path graph 3utility graph 4Reye graph

For an matrix, let denote any permutation , , ..., of the set of numbers 1, 2, ..., , and let be the character of the symmetric group corresponding to the partition . Then the immanant is defined aswhere the summation is over the permutations of the symmetric group and

A set partition of a set is a collection of disjoint subsets of whose union is . The number of partitions of the set is called a Bell number.

Let denote the partition lattice of the set . The maximum element of is(1)and the minimum element is(2)Let denote the number of chains of any length in containing both and . Then satisfies the recurrence relation(3)where and is a Stirling number of the second kind. The first few values of for , 2, ... are then 1, 1, 4, 32, 436, 9012, 262760, ... (OEIS A005121).Lengyel (1984) proved that the quotient(4)is bounded between two constants as , and Flajolet and Salvy (1990) improved the result of Babai and Lengyel (1992) to show that(5)(OEIS A086053).

Roman (1984, p. 26) defines "the" binomial identity as the equation(1)Iff the sequence satisfies this identity for all in a field of field characteristic 0, then is an associated sequence known as a binomial-type sequence.In general, a binomial identity is a formula expressing products of factors as a sum over terms, each including a binomial coefficient . The prototypical example is the binomial theorem(2)for . Abel (1826) gave a host of such identities (Riordan 1979, Roman 1984), some of which include(3)(4)(Abel 1826, Riordan 1979, p. 18; Roman 1984, pp. 30 and 73), and(5)(Saslaw 1989).

A Hadamard matrix is a type of square (-1,1)-matrix invented by Sylvester (1867) under the name of anallagmatic pavement, 26 years before Hadamard (1893) considered them. In a Hadamard matrix, placing any two columns or rows side by side gives half the adjacent cells the same sign and half the other sign. When viewed as pavements, cells with 1s are colored black and those with s are colored white. Therefore, the Hadamard matrix must have white squares (s) and black squares (1s).A Hadamard matrix of order is a solution to Hadamard's maximum determinant problem, i.e., has the maximum possible determinant (in absolute value) of any complex matrix with elements (Brenner and Cummings 1972), namely . An equivalent definition of the Hadamard matrices is given by(1)where is the identity matrix.A Hadamard matrix of order corresponds to a Hadamard design (, , ), and a Hadamard matrix gives a graph on vertices known as a Hadamard graphA complete set of Walsh..

A combinatorial conjecture formulated by Kneser (1955). It states that whenever the -subsets of a -set are divided into classes, then two disjoint subsets end up in the same class.Lovász (1978) gave a proof based on graph theory. In particular, he showed that the Kneser graph, whose vertices represent the -subsets, and where each edge connects two disjoint subsets, is not -colorable. More precisely, his results says that the chromatic number is equal to , and this implies that Kneser's conjecture is always false if the number of classes is increased to .An alternate proof was given by Bárány (1978).

A rook polynomial is a polynomial(1)whose number of ways nonattacking rooks can be arranged on an chessboard. The rook polynomials are given by(2)where is an associated Laguerre polynomial.The first few rook polynomials on square boards are(3)(4)(5)(6)(OEIS A021010).As an illustration, note that the case has two ways to place two rooks (i.e., the rook number ), four ways to place one rook (), and one way to place no rooks (), hence .

The numerators and denominators obtained by taking the ratios of adjacent terms in the triangular array of the number of "bordered" alternating sign matrices with a 1 at the top of column are, respectively, the numbers in the (2, 1)- and (1, 2)-Pascal triangles which are different from 1. This conjecture was proven by Zeilberger (1996).

The rascal triangle is a number triangle withnumbers arranged in staggered rows such that(1)The published study of this triangle seems to have originated relatively recently, having been added to Sloane's Online Encyclopedia of Integer Sequences (OEIS) as recently as 2002--where it was cataloged as [t]riangle with diagonal congruent to mod --and having been the subject of scholarly mathematical publication as recently as 2010 (Anggoro et al. 2010). The triangle is sometimes written without capitalization as the rascal triangle.One common point of exposition among literature regarding the rascal triangle is its similarity to Pascal's triangle. Indeed, the rascal triangle is topically similar to Pascal's triangle in that the configuration starting with begins(2)and that the rows afterwards have their first and last entries equal to(3)respectively.The similarities don't end there, however. One well-known fact about Pascal's..

A polygonal diagonal is a line segment connecting two nonadjacent polygon vertices of a polygon. The number of ways a fixed convex -gon can be divided into triangles by nonintersecting diagonals is (with diagonals), where is a Catalan number. This is Euler's polygon division problem. Counting the number of regions determined by drawing the diagonals of a regular -gon is a more difficult problem, as is determining the number of -tuples of concurrent diagonals (Kok 1972).The number of regions which the diagonals of a convexpolygon divide its center if no three are concurrent in its interior is(1)(2)The first few values are 0, 0, 1, 4, 11, 25, 50, 91, 154, 246, ... (OEIS A006522).

An alternating sign matrix is a matrix of 0s, 1s, and s in which the entries in each row or column sum to 1 and the nonzero entries in each row and column alternate in sign. The first few for , 2, ... are shown below:(1)(2)(3)(4)Such matrices satisfy the additional property that s in a row or column must have a "outside" it (i.e., all s are "bordered" by s). The numbers of alternating sign matrices for , 2, ... are given by 1, 2, 7, 42, 429, 7436, 218348, ... (OEIS A005130).The conjecture that the number of is explicitly given by the formula(5)now proven to be true, was known as the alternating sign matrix conjecture. can be expressed in closed form as a complicated function of Barnes G-functions, but additional simplification is likely possible.A recurrence relation for is given by(6)where is the gamma function.Let be the number of alternating sign matrices with one in the top row occurring in the th position. Then(7)The result(8)for..

The Euler numbers, also called the secant numbers or zig numbers, are defined for by(1)(2)where is the hyperbolic secant and sec is the secant. Euler numbers give the number of odd alternating permutations and are related to Genocchi numbers. The base e of the natural logarithm is sometimes known as Euler's number.A different sort of Euler number, the Euler number of a finite complex , is defined by(3)This Euler number is a topological invariant.To confuse matters further, the Euler characteristic is sometimes also called the "Euler number" and numbers produced by the prime-generating polynomial are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47). In this work, primes generated by that polynomial are termed Euler primes.Some values of the (secant) Euler numbers are(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(OEIS A000364).The slightly different convention defined by(16)(17)is..

Let there be integers with . The values represent the denominations of different coins, where these denominations have greatest common divisor of 1. The sums of money that can be represented using the given coins are then given by(1)where the are nonnegative integers giving the numbers of each coin used. If , it is obviously possibly to represent any quantity of money . However, in the general case, only some quantities can be produced. For example, if the allowed coins are , it is impossible to represent and 3, although all other quantities can be represented.Determining the function giving the greatest for which there is no solution is called the coin problem, or sometimes the money-changing problem. The largest such for a given problem is called the Frobenius number .The result(2)(3)(Nijenhuis and Wilf 1972) is mathematical folklore. The total number of such nonrepresentable amounts is given by(4)The largest nonrepresentable amounts for..

The Frobenius number is the largest value for which the Frobenius equation(1)has no solution, where the are positive integers, is an integer, and the solutions are nonnegative integer. As an example, if the values are 4 and 9, then 23 is the largest unsolvable number. Similarly, the largest number that is not a McNugget number (a number obtainable by adding multiples of 6, 9, and 20) is 43.Finding the Frobenius number of a given problem is known as the coinproblem.Computation of the Frobenius number is implemented in the Wolfram Language as FrobeniusNumber[a1, ..., an].Sylvester (1884) showed(2)(3)

Let be the area of a simply closed lattice polygon. Let denote the number of lattice points on the polygon edges and the number of points in the interior of the polygon. ThenThe formula has been generalized to three- and higherdimensions using Ehrhart polynomials.

A magic hexagon of order is an arrangement of close-packed hexagons containing the numbers 1, 2, ..., , where is the th hex number such that the numbers along each straight line add up to the same sum. (Here, the hex numbers are i.e., 1, 7, 19, 37, 61, 91, 127, ...; OEIS A003215). In the above magic hexagon of order , each line (those of lengths 3, 4, and 5) adds up to 38.It was discovered independently by Ernst von Haselberg in 1887 (Bauch 1990, Hemme 1990), W. Radcliffe in 1895 (Tapson 1987, Hemme 1990, Heinz), H. Lulli (Hendricks, Heinz), Martin Kühl in 1940 (Gardner 1963, 1984; Honsberger 1973), Clifford W. Adams, who worked on the problem from 1910 to 1957 (Gardner 1963, 1984; Honsberger 1973), and Vickers (1958; Trigg 1964).This problem and the solution have a long history. Adams came across the problem in 1910. He worked on the problem by trial and error and after many years arrived at the solution which he transmitted to M. Gardner,..

A one-person game played on a rectangular lattice of lamps which can be turned on and off. A move consists of flipping a "switch" inside one of the squares, thereby toggling the on/off state of this and all four vertically and horizontally adjacent squares. Starting from a randomly chosen light pattern, the aim is to turn all the lamps off. The problem of determining if it is possible to start from set of all lights being on to all lights being off is known as the "all-ones problem." As shown by Sutner (1989), this is always possible for a square lattice (Rangel-Mondragon).This can be translated into the following algebraic problem. 1. Each lamp configuration can be viewed as a matrix with entries in (i.e., a (0,1)-matrix, where each 1 represents a burning light and 0 represents a light turned off. For example, for the case,(1)2. The action of the switch placed at can be interpreted as the matrix addition , where is the matrix in which..

Given a (0,1)-matrix, fill 11 spaces in each row in such a way that all columns also have 11 spaces filled. Furthermore, each pair of rows must have exactly one filled space in the same column. This problem is equivalent to finding a projective plane of order 10. Using a computer program, Lam et al. (1989) showed that no such arrangement exists.Lam's problem is equivalent to finding nine orthogonal Latinsquares of order 10.

The problem of determining how many nonattacking knights can be placed on an chessboard. For , the solution is 32 (illustrated above). In general, the solutions are(1)giving the sequence 1, 4, 5, 8, 13, 18, 25, ... (OEIS A030978,Dudeney 1970, p. 96; Madachy 1979).The minimal number of knights needed to occupy or attack every square on an chessboard (i.e., domination numbers for the knight graphs) are given for , 2, ... by 1, 4, 4, 4, 5, 8, 10, 12, 14, ... (OEIS A006075), with corresponding numbers of such solutions given by 1, 1, 2, 3, 8, 22, 3, ... (OEIS A006076).

If is an integer, then for every residue class (mod ), there are infinitely many nonnegative integers for which , where is the partition function P.

Consider a set of positive integer-denomination postage stamps sorted such that . Suppose they are to be used on an envelope with room for no more than stamps. The postage stamp problem then consists of determining the smallest integer which cannot be represented by a linear combination with and .Without the latter restriction, this problem is known as the Frobenius problem or Frobenius postage stamp problem.The number of consecutive possible postage amounts is given by(1)where is called an -range.Exact solutions exist for arbitrary for and 3. The solution is(2)for . It is also known that(3)(Stöhr 1955, Guy 1994), where is the floor function, the first few values of which are 2, 4, 7, 10, 14, 18, 23, 28, 34, 40, ... (OEIS A014616; Guy 1994, p. 123).Hofmeister (1968, 1983) showed that for ,(4)where and are functions of (mod 9), and Mossige (1981, 1987) showed that(5)(Guy 1994, p. 123).Shallit (2002) proved that the (local) postage..

The symmetric group of degree is the group of all permutations on symbols. is therefore a permutation group of order and contains as subgroups every group of order .The th symmetric group is represented in the Wolfram Language as SymmetricGroup[n]. Its cycle index can be generated in the Wolfram Language using CycleIndexPolynomial[SymmetricGroup[n], x1, ..., xn]. The number of conjugacy classes of is given , where is the partition function P of . The symmetric group is a transitive group (Holton and Sheehan 1993, p. 27).For any finite group , Cayley's group theorem proves is isomorphic to a subgroup of a symmetric group.The multiplication table for is illustrated above.Let be the usual permutation cycle notation for a given permutation. Then the following table gives the multiplication table for , which has elements.(1)(2)(3)(1)(23)(3)(12)(123)(132)(2)(13)(1)(2)(3)(1)(2)(3)(1)(23)(3)(12)(123)(132)(2)(13)(1)(23)(1)(23)(1)(2)(3)(132)(2)(13)(3)(12)(123)(3)(12)(3)(12)(123)(1)(2)(3)(1)(23)(2)(13)(132)(123)(123)(3)(12)(2)(13)(132)(1)(2)(3)(1)(23)(132)(132)(2)(13)(1)(23)(1)(2)(3)(123)(3)(12)(2)(13)(2)(13)(132)(123)(3)(12)(1)(23)(1)(2)(3)This..

A permutation group is a finite group whose elements are permutations of a given set and whose group operation is composition of permutations in . Permutation groups have orders dividing .Two permutations form a group only if one is the identity element and the other is a permutation involution, i.e., a permutation which is its own inverse (Skiena 1990, p. 20). Every permutation group with more than two elements can be written as a product of transpositions.Permutation groups are represented in the Wolfram Language as a set of permutation cycles with PermutationGroup. A set of permutations may be tested to see if it forms a permutation group using PermutationGroupQ[l] in the Wolfram Language package Combinatorica` .Conjugacy classes of elements which are interchangedin a permutation group are called permutation cycles.Examples of permutation groups include the symmetric group (of order ), the alternating group (of order for ),..

In his Meditationes algebraicae, Waring (1770, 1782) proposed a generalization of Lagrange's four-square theorem, stating that every rational integer is the sum of a fixed number of th powers of positive integers, where is any given positive integer and depends only on . Waring originally speculated that , , and . In 1909, Hilbert proved the general conjecture using an identity in 25-fold multiple integrals (Rademacher and Toeplitz 1957, pp. 52-61).In Lagrange's four-square theorem, Lagrange proved that , where 4 may be reduced to 3 except for numbers of the form (as proved by Legendre; Hardy 1999, p. 12). In 1909, Wieferich proved that . In 1859, Liouville proved (using Lagrange's four-square theorem and Liouville polynomial identity) that . Hardy, and Little established , and this was subsequently reduced to by Balasubramanian et al. (1986). For the case , in 1896, Maillet began with a proof that , in 1909 Wieferich proved , and..

Consider the number of sequences that can be formed from permutations of a set of elements such that each partial sum is nonnegative. The number of sequences with nonnegative partial sums which can be formed from the permutations of 1s and s (Bailey 1996, Brualdi 1997) is given by the Catalan numbers . For example, the permutations of having nonnegative partial sums are , , , , and (1, , 1, , 1, ).Similarly, the number of nonnegative partial sums of 1s and s (Bailey 1996) is given bywhere these coefficients form Catalan's triangle(OEIS A009766) and

There are several closely related results that are variously known as the binomial theorem depending on the source. Even more confusingly a number of these (and other) related results are variously known as the binomial formula, binomial expansion, and binomial identity, and the identity itself is sometimes simply called the "binomial series" rather than "binomial theorem."The most general case of the binomial theorem is the binomialseries identity(1)where is a binomial coefficient and is a real number. This series converges for an integer, or . This general form is what Graham et al. (1994, p. 162). Arfken (1985, p. 307) calls the special case of this formula with the binomial theorem.When is a positive integer , the series terminates at and can be written in the form(2)This form of the identity is called the binomial theorem by Abramowitz and Stegun (1972, p. 10).The differing terminologies are..

There are several related series that are known as the binomial series.The most general is(1)where is a binomial coefficient and is a real number. This series converges for an integer, or (Graham et al. 1994, p. 162). When is a positive integer , the series terminates at and can be written in the form(2)The theorem that any one of these (or several other related forms) holds is knownas the binomial theorem.Special cases give the Taylor series(3)(4)where is a Pochhammer symbol and . Similarly,(5)(6)which is the so-called negative binomial series.In particular, the case gives(7)(8)(9)(OEIS A001790 and A046161), where is a double factorial and is a binomial coefficient.The binomial series has the continued fractionrepresentation(10)(Wall 1948, p. 343).

The fraction of odd values of the partition function P(n) is roughly 50%, independent of , whereas odd values of occur with ever decreasing frequency as becomes large. Kolberg (1959) proved that there are infinitely many even and odd values of .Leibniz noted that is prime for , 3, 4, 5, 6, but not 7. In fact, values of for which is prime are 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ... (OEIS A046063), corresponding to 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575). Numbers which cannot be written as a product of are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (OEIS A046064), corresponding to numbers of nonisomorphic Abelian groups which are not possible for any group order.Ramanujan conjectured a number of amazing and unexpected congruences involving . In particular, he proved(1)using Ramanujan's identity (Darling 1919; Hardy and Wright 1979; Drost 1997; Hardy 1999, pp. 87-88; Hirschhorn 1999). Ramanujan (1919) also showed that(2)and..

The factorial is defined for a positive integer as(1)So, for example, . An older notation for the factorial was written (Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978; Conway and Guy 1996).The special case is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).The factorial is implemented in the Wolfram Language as Factorial[n] or n!.The triangular number can be regarded as the additive analog of the factorial . Another relationship between factorials and triangular numbers is given by the identity(2)(K. MacMillan, pers. comm., Jan. 21, 2008).The factorial gives the number of ways in which objects can be permuted. For example, , since the six possible permutations of are , , , , , . The first few factorials for , 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (OEIS A000142).The..

Find the array of single digits which contains the maximum possible number of primes, where allowable primes may lie along any horizontal, vertical, or diagonal line.For the array, 11 primes are maximal and are contained in the two distinct arrays(1)giving the primes (3, 7, 13, 17, 31, 37, 41, 43,47, 71, 73) and (3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97), respectively.The best array is(2)which contains 30 primes: 3, 5, 7, 11, 13, 17, 31, 37, 41, 43, 47, 53, 59, 71, 73, 79, 97, 113, 157, 179, ... (OEIS A032529). This array was found by Rivera and Ayala and shown by Weisstein in May 1999 to be maximal and unique (modulo reflection and rotation).The best arrays known are(3)all of which contain 63 primes. The first was found by C. Rivera and J. Ayala in 1998, and the other three by James Bonfield on April 13, 1999. Mike Oakes proved by computation that the 63 primes is optimal for the array.The best prime arrays known are(4)each of which contains 116 primes...

It is possible to find six points in the plane, no three on a line and no four on a circle (i.e., none of which are collinear or concyclic), such that all the mutual distances are rational. An example is illustrated by Guy (1994, p. 185).It is not known if a triangle with integer sides, triangle medians, and area exists (although there are incorrect proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have showed that there are infinitely many triangles with rational sides (Heronian triangles) with two rational triangle medians (Guy 1994, p. 188).

A family of nonempty subsets of whose union contains the given set (and which contains no duplicated subsets) is called a cover (or covering) of . For example, there is only a single cover of , namely . However, there are five covers of , namely , , , , and .A minimal cover is a cover for which removal of one member destroys the covering property. For example, of the five covers of , only and are minimal covers. There are various other types of specialized covers, including proper covers, antichain covers, -covers, and -covers (Macula 1994).The number of possible covers for a set of elements arethe first few of which are 1, 5, 109, 32297, 2147321017, 9223372023970362989, ...(OEIS A003465).

The Catalan numbers on nonnegative integers are a set of numbers that arise in tree enumeration problems of the type, "In how many ways can a regular -gon be divided into triangles if different orientations are counted separately?" (Euler's polygon division problem). The solution is the Catalan number (Pólya 1956; Dörrie 1965; Honsberger 1973; Borwein and Bailey 2003, pp. 21-22), as graphically illustrated above (Dickau).Catalan numbers are commonly denoted (Graham et al. 1994; Stanley 1999b, p. 219; Pemmaraju and Skiena 2003, p. 169; this work) or (Goulden and Jackson 1983, p. 111), and less commonly (van Lint and Wilson 1992, p. 136).Catalan numbers are implemented in the WolframLanguage as CatalanNumber[n].The first few Catalan numbers for , 2, ... are 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ... (OEIS A000108).Explicit formulas for include(1)(2)(3)(4)(5)(6)(7)where..

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