Fibonacci number program - Recursive snippet. int fib(int n) { if (n < 2) return n; else return fib(n-1) + fib(n-2); } Runs in Θ(F(n)) time, which is Ω(1.6n). Fibonacci number program - Iterative snippet. int fib(int n) { int first = 0, second = 1; while (n--) { int tmp = first+second; first = second; second = tmp; } return first; } Fibonacci number program - Shorter iteration. This ve ...

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Fibonacci number program, Fibonacci number program - Common Lisp, Fibonacci number program - Calculating fibonacci through Lucas' formula, Fibonacci number program - Haskell examples, Fibonacci number program - Lazy infinite list, Fibonacci number program - Perl examples, Fibonacci number program - One example, Fibonacci number program - Binary recursion snippet, Fibonacci number program - Binary recursion with special Perl caching snippet, Fibonacci number program - Iterative snippet, Fibonacci number program - Command line iterative, Fibonacci number program - PostScript example, Fibonacci number program - Iterative, Fibonacci number program - Stack recursion, Fibonacci number program - Python examples, Fibonacci number program - Recursion, Fibonacci number program - Generator, Fibonacci number program - Matrix equation, Fibonacci number program - Scheme examples, Fibonacci number program - Binary recursion snippet, Fibonacci number program - Tail-end recursive snippet, Fibonacci number program - Tail-end recursive snippet, Fibonacci number program - Display all snippet, Fibonacci number program - C/C++/Java example, Fibonacci number program - Recursive snippet, Fibonacci number program - Iterative snippet, Fibonacci number program - Shorter iteration, Fibonacci number program - Ada example, Fibonacci number program - Recursive snippet, Fibonacci number program - Iterative snippet, Fibonacci number program - MatLab example, Fibonacci number program - Recursive snippet, Fibonacci number program - Iterative snippet, Fibonacci number program - PHP scripting language example, Fibonacci number program - Contained snippet, Fibonacci number program - Ruby examples, Fibonacci number program - QBasic/Visual Basic examples, Fibonacci number program - J examples, Fibonacci number program - Double recursion, Fibonacci number program - Single recursion, Fibonacci number program - Iteration, Fibonacci number program - Power of phi, Fibonacci number program - Continued fraction, Fibonacci number program - Taylor series, Fibonacci number program - Sum of binomial coefficients, Fibonacci number program - Matrix power, Fibonacci number program - Operations in Q[√5] and Z[√5]

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A regular pentagram is the {5/2} star polygon. It is most easily drawn by drawing a regular pentagon, joining the corners with lines and erasing the original pentagon. You may also extend the sides of the pentagon until they meet, obtaining a bigger pentagram. The golden ratio, φ = (1+√5)/2 = 1.618…, satisfying plays an important role in regular pentagons and pentagrams. Each line is divided into several smaller segments, and if you divide the length of the longer segment with the shorter segment of any pair of segments ...

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Pentagram, Pentagram - Geometry, Pentagram - Some relevant trigonometric values, Pentagram - History, Pentagram - Pythagorean use, Pentagram - Christian use, Pentagram - Satanic use, Pentagram - Neopagan Use, Pentagram - Brigate Rosse, Pentagram - Flags, Pentagram - In literature, Pentagram - 3D

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While one cannot discern any pattern in the infinite continued fraction expansion of π, this is not true for e, the base of the natural logarithm: The numbers with periodic continued fraction expansion are precisely the solutions of quadratic equations with integer coefficients. For example, the golden ratio φ = [1; 1, 1, 1, 1, 1, ...] and √ 2 = [1; 2, 2, 2, 2, ...]. However, most irrational numbers do not have any periodic or regular behavior in ...

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Continued fraction, Continued fraction - Motivation, Continued fraction - Calculating continued fraction representations, Continued fraction - Notations for continued fractions, Continued fraction - Finite continued fractions, Continued fraction - Infinite continued fractions, Continued fraction - Some useful theorems, Continued fraction - Theorem 1, Continued fraction - Theorem 2, Continued fraction - Theorem 3, Continued fraction - Theorem 4, Continued fraction - Theorem 5, Continued fraction - Semiconvergents, Continued fraction - Best rational approximations, Continued fraction - The continued fraction expansion of π, Continued fraction - Other continued fraction expansions, Continued fraction - Pell's equation, Continued fraction - Continued fractions and chaos

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Studios supplement these movies with independent productions, made with small budgets and often independently of the studio corporation. Movies made in this manner typically emphasize high professional quality in terms of acting, directing, screenwriting, and other elements associated with production, and also upon creativity and innovation. These movies usually rely upon critical praise or niche marketing to garner an audience. Because of an independent film's low budgets, a successful independent film can have a high profit-to-cost ratio, ...

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Cinema of the United States, Cinema of the United States - History, Cinema of the United States - Early development, Cinema of the United States - Rise of Hollywood, Cinema of the United States - Golden Age of Hollywood, Cinema of the United States - Changing realities and television's rise, Cinema of the United States - The 'New Hollywood' or Post-classical cinema, Cinema of the United States - Rise of the blockbuster, Cinema of the United States - Independent film, Cinema of the United States - Rise of the home video market, Cinema of the United States - Notable figures in U.S. film, Cinema of the United States - Bibliography

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Studios supplement these movies with independent productions, made with small budgets and often independently of the studio corporation. Movies made in this manner typically emphasize high professional quality in terms of acting, directing, screenwriting, and other elements associated with production, and also upon creativity and innovation. These movies usually rely upon critical praise or niche marketing to garner an audience. Because of an independent film's low budgets, a successful independent film can have a high profit-to-cost ratio, ...

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Cinema of the United States, Cinema of the United States - History, Cinema of the United States - Early development, Cinema of the United States - Rise of Hollywood, Cinema of the United States - Golden Age of Hollywood, Cinema of the United States - Changing realities and television's rise, Cinema of the United States - The 'New Hollywood' or Post-classical cinema, Cinema of the United States - Blockbusters, Cinema of the United States - Independent film, Cinema of the United States - Rise of the home video market, Cinema of the United States - Notable figures in U.S. film, Cinema of the United States - Bibliography

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Metrology One key of the ancient symbol-language, which concealed and revealed certain aspects of the esoteric teachings. It is seen in Hebrew metrology and its connection with the numerical values of the Hebrew letters, some clues to which were discovered by Ralston Skinner, author of The Source of Measures.

A measure, apart from number, reduces itself to a unit of measurement. It is hard to imagine how such a unit could be conceived, defined, or preserved, apart from physical objects; so that it would not be very surprising to find that such units have been preserved in ancient masonry. A number of well-defined units, generally called cubits, have thus been found.

If metrology is taken to include ratios, pi, the golden section, and other such constants may be sought among the proportions of ancient architecture. Clearly if we know the unit used, the length or other dimensions of a building will give us a number; and so those who knew the units would have the clue to the secret numbers.

(See also: Metrology, Mysticism, Mysticism Dictionary)

Golden mean philosophy - Crete. The earliest representation of this idea in culture is probably in the mythological Cretan tale of Daedalus and Icarus. Daedalus, a famous artist of his time, built feathered wings for himself and his son so that they might escape the clutches of King Minos. Daedalus warns his son to "fly the middle course", between the sea spray and the sun's heat. Icarus did not heed his father; he flew up and up until the sun melted the wax of his wings, and ...

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Golden mean philosophy, Golden mean philosophy - History of the golden mean in philosophy, Golden mean philosophy - Crete, Golden mean philosophy - Delphi, Golden mean philosophy - Pythagoreans, Golden mean philosophy - Socrates, Golden mean philosophy - Plato, Golden mean philosophy - Aristotle, Golden mean philosophy - Quotations, Golden mean philosophy - Miscellanea, Golden mean philosophy - Bibliography

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Pentagram, Pentalpha The five-pointed star, or star pentagon, called pentalpha by Pythagoreans because its corners are like five (pente) alphas (A). It combines the two and the three, or the first even number and the first odd number after unity, representing therefore on the universal plane the union of cosmic substance with cosmic intellect.

As a union or unity of five elements it stands for the heavenly or macrocosmic man, and its five points correspond to the head and limbs of the human body; the same general idea lies behind the five wounds which Christians ascribe to the crucified Jesus. Sometimes the five-pointed star is drawn with a point down and two horns up, signifying the polar opposite of the preceding, the nether or material pole of cosmic life, an emblem of matter and black magic. The decad is produced by a combination of these two; and thus we may obtain a still more profound emblem of man's dual nature.

It is likewise used for the number five, and thus represents the five root-races which have so far been manifested, and their corresponding five elements. The numbers of pi are given the form of geometrical figures, among which the 5 is shown as the pentagram. The number five plays an important part in mensuration and the proportions of the regular polyhedra, giving rise to the ratio of the golden section.

Finally, in theosophic symbology the pentalpha is frequently employed as the emblem of the true ego, the higher manas or buddhi-manas. It is likewise one of the emblematic figures containing one of the keys to the correct calculation of time periods, whether these be cosmic or terrestrial. It is quite a mistake to suppose that accurate computations of time periods may be arrived at by the simple arithmetical use of the number seven, whether by division, multiplication, or by a simple addition or subtraction; all such time periods are calculable solely on the basis of a correct knowledge of the respective uses of the five, six, and one.

(See also: Pentagram, Pentalpha, Mysticism, Mysticism Dictionary, Occultism, Occultism Dictionary)

According to the statistics released by the Immigration Department, about 800,000 to 900,000 Mainland visitors travel to Hong Kong each month. The ratio of visitors under the scheme rose from 5% to 16.8% between August and October 2003, and to more than 30% in May 2004. Tourism is a reflection of the economy. The increase in the number of Mainland tourists greatly benefits the economy of Hong Kong and it also shows that the economy is on an upward track. Average occupancy rate across all categories of hotels and tourist guest houses i ...

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Individual Visit Scheme, Individual Visit Scheme - October 1 Golden Week, Individual Visit Scheme - Economic Impact, Individual Visit Scheme - Social Impact, Individual Visit Scheme - Timetable of implementation

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Temple of Solomon The building of this temple, according to the Bible, was first projected by King David, but on command of the Lord was not carried out by him because he had "shed much blood." David, however, assembled materials and workmen. To aid him in building the Temple, his son Solomon appealed to Hiram or Huram, King of Tyre, to send him a skillful artisan, and King Hiram sent Hiram Abif to Solomon, also workmen and additional supplies of timber.

According to the Biblical account the Temple was completely built, while according to Masonic tradition the building was left unfinished on account of the death of Hiram Abif. The temple after its completion retained its original splendor for only 33 years when the Egyptian King Shishak made war upon Rehoboam, Solomon's son, captured Jerusalem, and took away all the treasures of the temple and of the king's house. Its history is one of repeated profanation and of alternate spoilations and repairs, until finally in 588 BC it was entirely destroyed by Nebuchadnezzar in the reign of Zedekiah. Yet Herodotus who, some 150 years later, visited Tyre and described the temple of Melkarth and Astoreth, does not even mention the Temple of Solomon, supporting the view that there never was such a structure actually built.

Granting that there may be some historical background for the Biblical account, it is nevertheless allegorical throughout. Blavatsky compares the measurements given in the Bible with those of the Great Pyramid and the Tabernacle of Moses, all of which were constructed upon the same abstract formula derived from the number of years in the precessional cycle, and also upon integral values of pi, the ratio of the circumference of a circle to the diameter. Moses symbolized these "under the form and measurements of the tabernacle, that he is supposed to have constructed in the wilderness. On these data the later Jewish High Priests constructed the allegory of Solomon's Temple -- a building which never had a real existence, any more than had King Solomon himself, who is simply, and as much a solar myth as is the still later Hiram Abif, of the Masons, as Ragon has well demonstrated. Thus, if the measurements of this allegorical temple, the symbol of the cycle of Initiation, coincide with those of the Great Pyramid, it is due to the fact that the former were derived from the latter through the Tabernacle of Moses" (SD 1:314-5). And she refers to "the undeniable, clear, and mathematical proofs that the esoteric foundations, or the system used in the building of the Great Pyramid, and the architectural measurements in the Temple of Solomon (whether the latter be mythical or real), Noah's ark, and the ark of the Covenant, are the same" (SD 2:465).

The key to the meaning of Solomon's Temple is given by W. Q. Judge: it

"means man whose frame is built up, finished and decorated without the least noise. But the materials had to be found, gathered together and fashioned in other and distant places. . . . Man could not have his bodily temple to live in until all the matter in and about his world had been found by the Master, who is the inner man, when found the plans for working it required to be detailed. They then had to be carried out in different detail until all the parts should be perfectly ready and fit for placing in the final structure. So in the vast stretch of time which began after the first almost intangible matter had been gathered and kneaded, the material and vegetable kingdoms had sole possession here with the Master -- man -- who was hidden from sight within carrying forward the plans for the foundations of the human temple. All of this requires many, many ages, since we know that nature never leaps. And when the rough work was completed, when the human temple was erected, many more ages would be required for all the servants, the priests, and the counselors to learn their parts properly so that man, the Master, might be able to use the temple for its best and highest purposes" (Ocean 20).

Thus David, who collected materials for the building but was not permitted actually to build the temple, represents the evolutionary and preparatory work of earlier rounds and of the earlier root-races preceding the middle of the third root-race of this round, when humanity appeared upon the scene -- Solomon, David's son -- takes up the task of the actual building of the human temple. David thus mystically may stand for the lunar or barhishad-pitris, and Solomon for the solar or agnishvatta-pitris.

According to the Old Testament, the building of the temple was completed, but it was used for its high purposes only briefly. Allegorically this was during the Golden Age of the childhood of the human race -- the building was complete only as regards childhood when the gods walked among mankind and were their divine instructors; but humanity was not yet truly human, for manas (mind) had not yet been awakened by the manasaputras of whom Hiram Abif was a type. It is here that Masonic tradition should be studied together with the Biblical account. Then with the awakening of manas, and the eating from the Tree of Knowledge and hence the power to choose between good and evil -- in other words, with the beginning of self-directed evolution, the temple was desecrated again and again. "The building of the Temple of Solomon is the symbolical representation of the gradual acquirement of the secret wisdom, or magic; the erection and development of the spiritual from the earthly; the manifestation of the power and splendor of the spirit in the physical world, through the wisdom and genius of the builder. The latter, when he has become an adept, is a mightier king than Solomon himself, the emblem of the sun or Light himself -- the light of the real subjective world, shining in the darkness of the objective universe. This is the 'Temple' which can be reared without the sound of the hammer, or any tool of iron being heard in the house while it is 'in building' " (IU 2:391).

Again, the building of a temple, sanctuary, Holy of Holies, etc., always signified in the occult language of ancient days the founding and dissemination throughout the world or a portion of mankind of a secret doctrine of nature. In a more restricted sense, the building of a temple referred to the actual establishment of an initiation center, where not only for such territory the ancient wisdom and its divine significances were taught, but disciples were trained and brought to the "new" or "second" birth, and thenceforth themselves became adepts or initiates. On these lines the building of Solomon's Temple was the inauguration and establishment of the teaching of nature's occult wisdom in Judea and surrounding territory.

(See also: Temple of Solomon, Mysticism, Mysticism Dictionary, Body mind and Soul)

Silver ratio - Definition as . The silver ratio (δS) is defined as the irrational number formed from the sum of 1 and the square root of 2. That is: It follows from this definition that: (δS − 1)2 = 2. Silver ratio - Definition as . The silver ratio can also be defined by the simple continued fract ...

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Silver ratio, Silver ratio - Definition, Silver ratio - Definition as, Silver ratio - Definition as, Silver ratio - Properties, Silver ratio - Silver means, Silver ratio - External link

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In diophantine approximation, the sequence of fractional parts of xn, n = 1, 2, 3, ... is shown to be equidistributed mod 1, for almost all real numbers x > 1. The silver ratio is an exception. ...

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Silver ratio, Silver ratio - Definition, Silver ratio - Definition as, Silver ratio - Definition as, Silver ratio - Properties, Silver ratio - Silver means, Silver ratio - External link

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A nautilus shell when viewed from above. The same shell viewed from underneath. The shell of the nautilus makes it as invisible as possible in the water. From the top looking down, the shell coloration blends in with the bottom or darkness below. When viewed from underneath, the shell is almost completely white, blending in with the light from above. ...

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Nautilus, Nautilus - Camouflage, Nautilus - Classification

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The Padovan sequence numbers can be written in terms of powers of the roots of the equation This equation has 3 roots; one real root p (known as the plastic number) and two complex conjugate roots q and r. Given these three roots, the Padovan sequence analogue of the Fibonacci sequence Binet formula is 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. For lar ...

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Padovan sequence, Padovan sequence - Recurrence relations, Padovan sequence - Extension to negative parameters, Padovan sequence - Sums of terms, Padovan sequence - Other identities, Padovan sequence - Binet-like formula, Padovan sequence - Combinatorial interpretations, Padovan sequence - Generating function, Padovan sequence - Generalizations, Padovan sequence - Padovan prime, Padovan sequence - Padovan L-System, Padovan sequence - Padovan Cuboid Spiral

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The Perrin sequence numbers can be written in terms of powers of the roots of the equation x3 − x − 1 = 0. This equation has 3 roots; one real root p (known as the plastic number) and two complex conjugate roots q and r. Given these three roots, the Perrin sequence analogue of the Fibonacci sequence Binet formula is 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. For large ...

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Perrin number, Perrin number - History, Perrin number - Generating function, Perrin number - Matrix Form, Perrin number - Binet Like Formula, Perrin number - Perrin prime

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The icosidodecahedron is a rectified dodecahedron and also a rectified icosahedron. Compare: ...

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Icosidodecahedron, Icosidodecahedron - Related polyhedra

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If you blow up an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the faces, and do the same to its dual dodecahedron, and patch the square holes in the result, you get a rhombicosadodecahedron. Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron. The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" small rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is ...

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Rhombicosidodecahedron, Rhombicosidodecahedron - Canonical coordinates, Rhombicosidodecahedron - Geometric relations

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The English physician and alchemist Robert Fludd (1574-1637) expicitly based his work Utriusque Cosmi Historia (The history of the two worlds) upon the macro/micro correspondence; as does Sir Thomas Browne in his binary Discourses of 1658: Hydriotaphia, Urn Burial depicts the small, temporal world of man, whilst The Garden of Cyrus represents the macrocosm, in which the ubiquitous a ...

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Macrocosm and microcosm, Macrocosm and microcosm - Ancient thought, Macrocosm and microcosm - Medieval and modern thought, Macrocosm and microcosm - Bibliography

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See also the "Relationship to other functions" section below. For integer values of s, we have the following explicit expressions: The polylogarithm for all negative integer values of s can be expressed as a ratio of polynomials in z (See series representations below). Some particular expressions for half-integer values of the argument are: where ζ is the Riemann zeta function. No similar formulas of this type are known for higher orders See also:

Polylogarithm, Polylogarithm - Properties, Polylogarithm - Particular values, Polylogarithm - Alternate expressions, Polylogarithm - Relationship to other functions, Polylogarithm - Series representations, Polylogarithm - Limiting behavior, Polylogarithm - The dilogarithm, Polylogarithm - Polylogarithm ladders

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The Classical Orders are largely known through the writings of Vitruvius, particularly De Archetura (The Ten Books of Architecture) and studies of classical architecture by Renaissance architects and historians. Within a classical order elements from the positioning of triglyphs to the overall height and width of the building were controlled by principles of proportionality. See also:

Proportion architecture, Proportion architecture - Sacred Proportions, Proportion architecture - Feng Shui, Proportion architecture - Classical Orders, Proportion architecture - Vitruvian Proportion, Proportion architecture - Renaissance Orders, Proportion architecture - Le Modulor, Proportion architecture - The Plastic Number

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The generating function of the Padovan sequence is This can be used to prove identities involving products of the Padovan sequence with geometric terms, such as: ...

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Padovan sequence, Padovan sequence - Recurrence relations, Padovan sequence - Extension to negative parameters, Padovan sequence - Sums of terms, Padovan sequence - Other identities, Padovan sequence - Binet-like formula, Padovan sequence - Combinatorial interpretations, Padovan sequence - Generating function, Padovan sequence - Generalizations, Padovan sequence - Padovan prime, Padovan sequence - Padovan L-System, Padovan sequence - Padovan Cuboid Spiral

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Khinchin's constant may be expressed as a rational zeta series in the form or, by peeling off terms in the series, where N is an integer, held fixed, and ζ(s,n) is the Hurwitz zeta function. Both series are strongly convergent, as ζ(n) − 1 approaches zero quickly for large n. An expansion may also ...

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Khinchin's constant, Khinchin's constant - Series expressions, Khinchin's constant - Hölder means, Khinchin's constant - Harmonic mean

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