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Diamond

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...

Kings problem

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)

Desargues' theorem

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.

Hard hexagon entropy constant

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...

Projective plane

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)..

Nexus number

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, ...

Mill curve

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

Central fibonomial 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).

Star of david theorem

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..

Idempotent number

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

Bernoulli triangle

(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.

Fibonomial coefficient

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

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..

Composition

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..

Waring's problem

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..

Nonnegative partial sum

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

Binomial theorem

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..

Binomial series

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).

Partition function p congruences

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..

Factorial

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..

Prime array

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...

Rational distances

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).

Cover

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).

Catalan number

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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