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Alcuin's Sequence
In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of: : \frac = x^3 + x^5 + x^6 + 2x^7 + x^8 + 3x^9 + \cdots. The sequence begins with these integers: : 0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21 The ''n''th term is the number of triangles with integer sides and perimeter ''n''. It is also the number of triangles with ''distinct'' integer sides and perimeter ''n'' + 6, i.e. number of triples (''a'', ''b'', ''c'') such that 1 ≤ ''a'' < ''b'' < ''c'' < ''a'' + ''b'', ''a'' + ''b'' + ''c'' = ''n'' + 6. If one deletes the three leading zeros, then it is the number of ways in which ''n'' empty casks, ''n'' casks half-full of wine and ''n'' full casks can be distributed to three persons in such a way that each one gets the same number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alcuin Of York
Alcuin of York (; la, Flaccus Albinus Alcuinus; 735 – 19 May 804) – also called Ealhwine, Alhwin, or Alchoin – was a scholar, clergyman, poet, and teacher from York, Northumbria. He was born around 735 and became the student of Archbishop Ecgbert at York. At the invitation of Charlemagne, he became a leading scholar and teacher at the Carolingian court, where he remained a figure in the 780s and 790s. Before that, he was also a court chancellor in Aachen. "The most learned man anywhere to be found", according to Einhard's ''Life of Charlemagne'' (–833), he is considered among the most important intellectual architects of the Carolingian Renaissance. Among his pupils were many of the dominant intellectuals of the Carolingian era. During this period, he perfected Carolingian minuscule, an easily read manuscript hand using a mixture of upper- and lower-case letters. Latin paleography in the eighth century leaves little room for a single origin of the scr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heron Triangle
In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula. Heron's formula implies that the Heronian triangles are exactly the positive integer solutions of the Diophantine equation :16\,A^2=(a+b+c)(a+b-c)(b+c-a)(c+a-b); that is, the side lengths and area of any Heronian triangle satisfy the equation, and any positive integer solution of the equation describes a Heronian triangle. If the three side lengths are setwise coprime, the Heronian triangle is called ''primitive''. Triangles whose side lengths and areas are all rational numbers (positive rational solutions of the above equation) are sometimes also called ''Heronian triangles'' or rational triangles; in this article, these more general triangles will be called ''rational Heronian triangles''. Every (integral) Heronian triangle is a rational Heronian triangle. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositiones Ad Acuendos Juvenes
The medieval Latin manuscript ''Propositiones ad Acuendos Juvenes'' ( en, Problems to Sharpen the Young) is one of the earliest known collections of recreational mathematics problems. David Darling, ''The Internet Encyclopedia of Science''. Accessed on line February 7, 2008. The oldest known copy of the manuscript dates from the late 9th century. The text is attributed to (died 804.) Some editions of the text contain 53 problems, others 56. It has been translated into English by John Hadley, with annotations by John Hadley and . [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
