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Bernoulli's Triangle
Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer ''n'' and for any integer ''k'' included between 0 and ''n'', the component in row ''n'' and column ''k'' is given by: : \sum_^k , i.e., the sum of the first ''k'' ''n''th-order binomial coefficients. The first rows of Bernoulli's triangle are: : \begin & k & 0 & 1 & 2 & 3 & 4 & 5\\ n & & \\ \hline 0 & & 1 \\ 1 & & 1 & 2 \\ 2 & & 1 & 3 & 4 \\ 3 & & 1 & 4 & 7 & 8 \\ 4 & & 1 & 5 & 11 & 15 & 16 \\ 5 & & 1 & 6 & 16 & 26 & 31 & 32 \end Similarly to Pascal's triangle, each component of Bernoulli's triangle is the sum of two components of the previous row, except for the last number of each row, which is double the last number of the previous row. For example, if B_ denotes the component in row ''n'' and column ''k'', then: : \begin B_=&B_+B_ &\mbox&k [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Triangle Derivation
Bernoulli can refer to: People *Bernoulli family of 17th and 18th century Swiss mathematicians: **Daniel Bernoulli (1700–1782), developer of Bernoulli's principle **Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli numbers are named **Jacob II Bernoulli (1759–1789) **Johann Bernoulli (1667–1748) **Johann II Bernoulli (1710–1790) **Johann III Bernoulli (1744–1807), also known as Jean, astronomer **Nicolaus I Bernoulli (1687–1759) **Nicolaus II Bernoulli (1695–1726) * Elisabeth Bernoulli (1873–1935), Swiss temperance campaigner *Hans Benno Bernoulli (1876–1959), Swiss architect *Ludwig Bernoully (1873–1928), German architect Mathematics * Bernoulli differential equation * Bernoulli distribution and Bernoulli random variable * Bernoulli's inequality * Bernoulli's triangle * Bernoulli number * Bernoulli polynomials * Bernoulli process * Bernoulli trial * Lemniscate of Bernoulli Science * 2034 Bernoulli, minor planet * Bernoulli's principle, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Array
An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the sums of their horizontal segments form a succession of twelve-tone aggregates * Array mbira, a musical instrument * Spiral array model, a music pitch space Science Astronomy A telescope array, also called astronomical interferometer. Biology * Various kinds of multiple biological arrays called microarrays * Visual feature array, a model for the visual cortex Computer science Generally, a collection of same type data items that can be selected by indices computed at run-time, including: * Array data structure, an arrangement of items at equally spaced addresses in computer memory * Array data type, used in a programming language to specify a variable that can be indexed * Associative array, an abstract data structure model comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Sums
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal Triangle Compositions
Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French mathematician, physicist, inventor, philosopher, writer and theologian Places * Pascal (crater), a lunar crater * Pascal Island (Antarctica) * Pascal Island (Western Australia) Science and technology * Pascal (unit), the SI unit of pressure * Pascal (programming language), a programming language developed by Niklaus Wirth * PASCAL (database), a bibliographic database maintained by the Institute of Scientific and Technical Information * Pascal (microarchitecture), codename for a microarchitecture developed by Nvidia Other uses * (1895–1911) * (1931–1942) * Pascal and Maximus, fictional characters in ''Tangled'' * Pascal blanc, a French white wine grape * Pascal College, secondary education school in Zaandam, the Netherlands * Pasca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Triangle Columns
Bernoulli can refer to: People *Bernoulli family of 17th and 18th century Swiss mathematicians: **Daniel Bernoulli (1700–1782), developer of Bernoulli's principle **Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli numbers are named **Jacob II Bernoulli (1759–1789) **Johann Bernoulli (1667–1748) **Johann II Bernoulli (1710–1790) **Johann III Bernoulli (1744–1807), also known as Jean, astronomer **Nicolaus I Bernoulli (1687–1759) **Nicolaus II Bernoulli (1695–1726) * Elisabeth Bernoulli (1873–1935), Swiss temperance campaigner *Hans Benno Bernoulli (1876–1959), Swiss architect *Ludwig Bernoully (1873–1928), German architect Mathematics * Bernoulli differential equation * Bernoulli distribution and Bernoulli random variable * Bernoulli's inequality * Bernoulli's triangle * Bernoulli number * Bernoulli polynomials * Bernoulli process * Bernoulli trial * Lemniscate of Bernoulli Science * 2034 Bernoulli, minor planet * Bernoulli's principle, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Applications of Fibonacci numbers include co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazy Caterer's Sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a disk (a pancake or pizza is usually used to describe the situation) that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point inside the circle, but up to seven if they do not. This problem can be formalized mathematically as one of counting the cells in an arrangement of lines; for generalizations to higher dimensions, ''see'' arrangement of hyperplanes. The analogue of this sequence in three dimensions is the cake number. Formula and sequence The maximum number ''p'' of pieces that can be created with a given number of cuts , where , is given by the formula : p = \frac. Using binomial coefficients, the formula can be expressed as :p = 1 + \dbinom = \dbinom+\dbinom+\dbinom. Simply put, each number equals a triangular number plus 1. As the third col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cake Number
In mathematics, the cake number, denoted by ''Cn'', is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly ''n'' planes. The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence. The values of ''Cn'' for increasing are given by General formula If ''n''! denotes the factorial, and we denote the binomial coefficients by : = \frac , and we assume that ''n'' planes are available to partition the cube, then the ''n''-th cake number is: : C_n = + + + = \tfrac\left(n^3 + 5n + 6\right) = \tfrac\left(n+1) (n (n-1) + 6\right). Properties The only cake number which is prime is 2, since it requires \left(n+1) (n (n-1) + 6\right) to have prime factorisation 2 \cdot 3 \cdot p where p is some prime. This is impossible for n > 2 as we know \left(n (n-1) + 6\right) must be even, so it must be e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dividing A Circle Into Areas
The number of and for first 6 terms of Moser's circle problem In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with ''n'' sides in such a way as to ''maximise'' the number of areas created by the edges and diagonals, sometimes called Moser's circle problem, has a solution by an inductive method. The greatest possible number of regions, , giving the sequence 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (). Though the first five terms match the geometric progression , it diverges at , showing the risk of generalising from only a few observations. Lemma If there are ''n'' points on the circle and one more point is added, ''n'' lines can be drawn from the new point to previously existing points. Two cases are possible. In the first case (a), the new line passes through a point where two or more old lines (between previously existing points) cross. In the second case (b), the new line crosses each of the old lines in a different point. It wil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space (mathematics)
In mathematics, a space is a set (sometimes called a universe) with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can be elements of a set, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space. More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional Euclidean space are uniquely determined by Euclid's axioms, and all three-dimensional Euclidean spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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