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54-619: The newly refurbished Unna station has trains to all major cities in North Rhine Westphalia including Dortmund , Cologne , Münster , Hamm , Düsseldorf and Wuppertal . There is also the Regional-Express 7 ( Rhein-Münsterland-Express ), which runs from Rheine via Cologne to Krefeld . Unna is situated on an ancient salt-trading route, the Westphalian Hellweg . Trade on this route and during

108-748: A φ n + b ψ n {\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}} where a = U 1 − U 0 ψ 5 , b = U 0 φ − U 1 5 . {\displaystyle {\begin{aligned}a&={\frac {U_{1}-U_{0}\psi }{\sqrt {5}}},\\[3mu]b&={\frac {U_{0}\varphi -U_{1}}{\sqrt {5}}}.\end{aligned}}} Since | ψ n 5 | < 1 2 {\textstyle \left|{\frac {\psi ^{n}}{\sqrt {5}}}\right|<{\frac {1}{2}}} for all n ≥ 0 ,

162-589: A φ n − 1 + b ψ n − 1 + a φ n − 2 + b ψ n − 2 = U n − 1 + U n − 2 . {\displaystyle {\begin{aligned}U_{n}&=a\varphi ^{n}+b\psi ^{n}\\[3mu]&=a(\varphi ^{n-1}+\varphi ^{n-2})+b(\psi ^{n-1}+\psi ^{n-2})\\[3mu]&=a\varphi ^{n-1}+b\psi ^{n-1}+a\varphi ^{n-2}+b\psi ^{n-2}\\[3mu]&=U_{n-1}+U_{n-2}.\end{aligned}}} If

216-402: A = 1 φ − ψ = 1 5 , b = − a , {\displaystyle a={\frac {1}{\varphi -\psi }}={\frac {1}{\sqrt {5}}},\quad b=-a,} producing the required formula. Taking the starting values U 0 and U 1 to be arbitrary constants, a more general solution is: U n =

270-539: A and b are chosen so that U 0 = 0 and U 1 = 1 then the resulting sequence U n must be the Fibonacci sequence. This is the same as requiring a and b satisfy the system of equations: { a + b = 0 φ a + ψ b = 1 {\displaystyle \left\{{\begin{aligned}a+b&=0\\\varphi a+\psi b&=1\end{aligned}}\right.} which has solution

324-605: A and b , the sequence defined by U n = a φ n + b ψ n {\displaystyle U_{n}=a\varphi ^{n}+b\psi ^{n}} satisfies the same recurrence, U n = a φ n + b ψ n = a ( φ n − 1 + φ n − 2 ) + b ( ψ n − 1 + ψ n − 2 ) =

378-545: A city festival are located in the Old Market Square, stretching from there through the pedestrian area to the city hall. Unna is home to a large community of artists, some of whose works are on public display in the city. In the Old Market Square, e.g., there is a statue by painter and sculptor Josef Baron , depicting a man pulling a stubborn donkey, which is the city mascot. A common part of traditional German drinking culture, numerous breweries once formed part of

432-438: A field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the rabbit math problem : how many pairs will there be in one year? At the end of the n -th month, the number of pairs of rabbits is equal to the number of mature pairs (that is, the number of pairs in month n – 2 ) plus

486-417: A given total duration results in the Fibonacci numbers: the number of patterns of duration m units is F m +1 . Knowledge of the Fibonacci sequence was expressed as early as Pingala ( c.  450 BC–200 BC). Singh cites Pingala's cryptic formula misrau cha ("the two are mixed") and scholars who interpret it in context as saying that the number of patterns for m beats ( F m +1 )

540-401: A stem , the fruit sprouts of a pineapple , the flowering of an artichoke , and the arrangement of a pine cone 's bracts, though they do not occur in all species. Fibonacci numbers are also strongly related to the golden ratio : Binet's formula expresses the n -th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to

594-481: Is asymptotic to φ n / 5 {\displaystyle \varphi ^{n}/{\sqrt {5}}} , the number of digits in F n is asymptotic to n log 10 ⁡ φ ≈ 0.2090 n {\displaystyle n\log _{10}\varphi \approx 0.2090\,n} . As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits. More generally, in

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648-501: Is extended to negative integers using the Fibonacci rule F n = F n + 2 − F n + 1 . {\displaystyle F_{n}=F_{n+2}-F_{n+1}.} Binet's formula provides a proof that a positive integer x is a Fibonacci number if and only if at least one of 5 x 2 + 4 {\displaystyle 5x^{2}+4} or 5 x 2 − 4 {\displaystyle 5x^{2}-4}

702-486: Is a perfect square . This is because Binet's formula, which can be written as F n = ( φ n − ( − 1 ) n φ − n ) / 5 {\displaystyle F_{n}=(\varphi ^{n}-(-1)^{n}\varphi ^{-n})/{\sqrt {5}}} , can be multiplied by 5 φ n {\displaystyle {\sqrt {5}}\varphi ^{n}} and solved as

756-634: Is also a DPD distribution centre along with a central distribution depot for the pump producer Wilo . Another large logistics complex belonging to DHL came into operation in 2008. Unna is twinned with: Unna station Unna station is the main passenger station in the Westphalian city of Unna in the German state of North Rhine-Westphalia . The other stations in the city that are served by regular passenger services are Unna-Königsborn , Unna West , Massen , Lünern and Hemmerde. The station

810-421: Is also the case that ψ 2 = ψ + 1 {\displaystyle \psi ^{2}=\psi +1} and it is also the case that ψ n = F n ψ + F n − 1 . {\displaystyle \psi ^{n}=F_{n}\psi +F_{n-1}.} These expressions are also true for n < 1 if the Fibonacci sequence F n

864-544: Is an entire journal dedicated to their study, the Fibonacci Quarterly . Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure , and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings , such as branching in trees, the arrangement of leaves on

918-517: Is nearby. The River Ruhr runs just south of Unna through Fröndenberg , before heading through the main part of the Ruhr district. Unna consists of the following districts: Massen and Königsborn are former industrial and mining areas; the other districts have a more rural character. The history of human settlement in what is now the city of Unna can be traced back to the Neolithic Era. In

972-507: Is obtained by adding one [S] to the F m cases and one [L] to the F m −1 cases. Bharata Muni also expresses knowledge of the sequence in the Natya Shastra (c. 100 BC–c. 350 AD). However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135): Variations of two earlier meters [is

1026-1043: Is the golden ratio , and ψ is its conjugate : ψ = 1 − 5 2 = 1 − φ = − 1 φ ≈ − 0.61803 39887 … . {\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}=1-\varphi =-{1 \over \varphi }\approx -0.61803\,39887\ldots .} Since ψ = − φ − 1 {\displaystyle \psi =-\varphi ^{-1}} , this formula can also be written as F n = φ n − ( − φ ) − n 5 = φ n − ( − φ ) − n 2 φ − 1 . {\displaystyle F_{n}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{\sqrt {5}}}={\frac {\varphi ^{n}-(-\varphi )^{-n}}{2\varphi -1}}.} To see

1080-475: Is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio φ : {\displaystyle \varphi \colon } lim n → ∞ F n + 1 F n = φ . {\displaystyle \lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\varphi .} This convergence holds regardless of

1134-526: Is valid for n > 2 . The first 20 Fibonacci numbers F n are: The Fibonacci sequence appears in Indian mathematics , in connection with Sanskrit prosody . In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with

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1188-516: The Middle Ages , Unna gained significance as a way station on the Hellweg . It is first recorded by name in an ecclesiastical document of 1032. Around 1200, Count Friedrich von Altena-Isenberg was invested with the fiefdom of Unna, among other estates, by the archbishop-electorate of Cologne . Over the next few hundred years the town was repeatedly fought over, and burned down several times. In

1242-630: The Prussian state railways opened the Fröndenberg–Kamen railway to connect the three east-west lines in the area. The southern part connected with the line to Menden , which was opened in 1872; this line was extended in 1912 to Neuenrade as the Hönne Valley Railway . The northern part was opened later for freight in 1900 and a year later for passenger traffic. This route is now worked only as far as Unna-Königsborn, as line S 4 of

1296-878: The Rhine-Ruhr S-Bahn . Between 1909 and 1950, the Unna-Kamen-Werne light railway also connected Unna station and Kamen station . Unna station is served by the Rhein-Münsterland-Express (RE 7, at 60-minute intervals), the Maas-Wupper-Express (RE 13, at 60-minute intervals), the Hönnetalbahn (RB 54, at 60-minute intervals), the Hellweg-Bahn (RB 59, at 30-minute intervals) and Rhine-Ruhr S-Bahn line S 4 (at 30-minute intervals). The station also serves as

1350-426: The base b representation, the number of digits in F n is asymptotic to n log b ⁡ φ = n log ⁡ φ log ⁡ b . {\displaystyle n\log _{b}\varphi ={\frac {n\log \varphi }{\log b}}.} Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges . He wrote that "as 5

1404-1188: The floor function gives the largest index of a Fibonacci number that is not greater than F : n l a r g e s t ( F ) = ⌊ log φ ⁡ 5 ( F + 1 / 2 ) ⌋ ,   F ≥ 0 , {\displaystyle n_{\mathrm {largest} }(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}(F+1/2)\right\rfloor ,\ F\geq 0,} where log φ ⁡ ( x ) = ln ⁡ ( x ) / ln ⁡ ( φ ) = log 10 ⁡ ( x ) / log 10 ⁡ ( φ ) {\displaystyle \log _{\varphi }(x)=\ln(x)/\ln(\varphi )=\log _{10}(x)/\log _{10}(\varphi )} , ln ⁡ ( φ ) = 0.481211 … {\displaystyle \ln(\varphi )=0.481211\ldots } , and log 10 ⁡ ( φ ) = 0.208987 … {\displaystyle \log _{10}(\varphi )=0.208987\ldots } . Since F n

1458-539: The nickel alloys producer VDM Metals operates a melting and casting plant in Unna, for example. Also known is Alexis Tsiami, who has developed new methods for opening car doors. In recent years Unna has boomed as a logistics centre. It houses the former distribution centre for the (recently much diminished) Karstadt department store chain; the distribution centre has been acquired by the DHL division of Deutsche Post . There

1512-475: The 14th century the town became wealthy: a mint was established and regional trade blossomed. This is documented by the discovery of around 70 gold coins during excavation works in 1952. The coins originated from various countries and are thought to have been buried around 1375. From the mid-15th century on, the city was a notable trade centre and member of the Hanseatic League . In 1597 more than half

1566-475: The 19th century close to the heart of the city. Its landmark is an installation of Fibonacci numbers by Italian artist Mario Merz on the brewery's chimney (the thirteenth-century mathematician Leonardo Fibonacci lived in Pisa , twinned with Unna since 1996). The light art installations are integrated into the industrial structures of the brewery's former cellar vaults. The former brewery buildings are also home to

1620-537: The Fibonacci recursion. In other words, φ n = φ n − 1 + φ n − 2 , ψ n = ψ n − 1 + ψ n − 2 . {\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\[3mu]\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}} It follows that for any values

1674-473: The artisan district which had survived bombing was largely torn down to make way for modern development; however many of the buildings have been restored. Unna is seat of the world's only art museum dedicated exclusively to the collection and presentation of Light art , the Centre for International Light Art (CILA). It is located in the former Linden brewery, a red brick industrial building complex dating from

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1728-461: The central bus junction for the city. Fibonacci number In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers , commonly denoted F n   . Many writers begin the sequence with 0 and 1, although some authors start it from 1 and 1 and some (as did Fibonacci) from 1 and 2. Starting from 0 and 1,

1782-616: The city library, the adult education centre (Volkshochschule), and the tourist information centre. Unna's Hellweg Museum , a regional history museum, is located in the medieval Unna castle. Many historic buildings as well as parts of the city wall, including towers near the artisan quarter, remain intact and in good condition. Unna holds the largest Italian festival north of Italy every two years (happily named Un(n)a Festa Italiana ), when buildings are decorated with light installations by artists from Bari in Italy. An annual Christmas market and

1836-481: The city's Herting gate or 'port'. The liquor is produced near where the gate used to be, and is sold locally. Until the mid-nineteenth century the focus of Unna's economy was on the region's agriculture. Industrialisation rapidly followed. In contrast with the switch to service sector employment in some industrial towns further west in the Ruhr area, most of the jobs in Unna are still in heavy industry (iron and metal work, machine manufacturing) or craft based. Since 1972,

1890-578: The cityscape, of which the largest and most well known was the Lindenbrauerei (formerly Linden-Adler-Brauerei ). It marketed its products under the name Lindenbier or Lindenpils . Most of the breweries have since closed down, but the Lindenbrauerei started a small-scale production again at the beginning of the 21st century. In common with other German towns, Unna also produces its own traditional herbal liquor, 'Herting Pörter', named after

1944-630: The golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers , which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences . The Fibonacci numbers may be defined by the recurrence relation F 0 = 0 , F 1 = 1 , {\displaystyle F_{0}=0,\quad F_{1}=1,} and F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}} for n > 1 . Under some older definitions,

1998-546: The golden ratio satisfies the equation φ 2 = φ + 1 , {\displaystyle \varphi ^{2}=\varphi +1,} this expression can be used to decompose higher powers φ n {\displaystyle \varphi ^{n}} as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of φ {\displaystyle \varphi } and 1. The resulting recurrence relationships yield Fibonacci numbers as

2052-407: The last is the number ... of the next mātrā-vṛtta." The Fibonacci sequence first appears in the book Liber Abaci ( The Book of Calculation , 1202) by Fibonacci where it is used to calculate the growth of rabbit populations. Fibonacci considers the growth of an idealized ( biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in

2106-1129: The linear coefficients : φ n = F n φ + F n − 1 . {\displaystyle \varphi ^{n}=F_{n}\varphi +F_{n-1}.} This equation can be proved by induction on n ≥ 1 : φ n + 1 = ( F n φ + F n − 1 ) φ = F n φ 2 + F n − 1 φ = F n ( φ + 1 ) + F n − 1 φ = ( F n + F n − 1 ) φ + F n = F n + 1 φ + F n . {\displaystyle \varphi ^{n+1}=(F_{n}\varphi +F_{n-1})\varphi =F_{n}\varphi ^{2}+F_{n-1}\varphi =F_{n}(\varphi +1)+F_{n-1}\varphi =(F_{n}+F_{n-1})\varphi +F_{n}=F_{n+1}\varphi +F_{n}.} For ψ = − 1 / φ {\displaystyle \psi =-1/\varphi } , it

2160-518: The mineral springs could still be used for health purposes. During the Second World War , in 1943-45 there were major air attacks directed at the significant barracks and other military installations in the city. In the older part of the city, there are many half-timbered buildings built between the 16th and 19th centuries. Unna's economy was largely based on agriculture until the 19th century, when it became industrialised. After World War II,

2214-505: The number F n is the closest integer to φ n 5 {\displaystyle {\frac {\varphi ^{n}}{\sqrt {5}}}} . Therefore, it can be found by rounding , using the nearest integer function: F n = ⌊ φ n 5 ⌉ ,   n ≥ 0. {\displaystyle F_{n}=\left\lfloor {\frac {\varphi ^{n}}{\sqrt {5}}}\right\rceil ,\ n\geq 0.} In fact,

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2268-491: The number of pairs alive last month (month n – 1 ). The number in the n -th month is the n -th Fibonacci number. The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas . Like every sequence defined by a homogeneous linear recurrence with constant coefficients , the Fibonacci numbers have a closed-form expression . It has become known as Binet's formula , named after French mathematician Jacques Philippe Marie Binet , though it

2322-520: The period of the Hanseatic League came from as far as London. The city is located at the eastern extremity of the Ruhr district , about 15 kilometres (9 miles) east of the centre of Dortmund . Unna also serves as a dormitory city, being home to many commuters who work in Dortmund and other nearby cities. Local dialects of German include Westfälisch and Ruhrpott . The recreational district of Sauerland

2376-512: The population died of the Bubonic plague . In the early 17th century, the town changed hands several times in religious wars, and in 1666 fell under the control of Prussia . In the early 19th century, the primary character of the town started to change from agricultural to industrial, with improved communications by road, rail and waterways. Coal mining started in 1870, together with industries dependent on it. The population rose from around 2,500 at

2430-409: The relation between the sequence and these constants, note that φ and ψ are both solutions of the equation x 2 = x + 1 {\textstyle x^{2}=x+1} and thus x n = x n − 1 + x n − 2 , {\displaystyle x^{n}=x^{n-1}+x^{n-2},} so the powers of φ and ψ satisfy

2484-485: The rounding error quickly becomes very small as n grows, being less than 0.1 for n ≥ 4 , and less than 0.01 for n ≥ 8 . This formula is easily inverted to find an index of a Fibonacci number F : n ( F ) = ⌊ log φ ⁡ 5 F ⌉ ,   F ≥ 1. {\displaystyle n(F)=\left\lfloor \log _{\varphi }{\sqrt {5}}F\right\rceil ,\ F\geq 1.} Instead using

2538-427: The same convergence towards the golden ratio. In general, lim n → ∞ F n + m F n = φ m {\displaystyle \lim _{n\to \infty }{\frac {F_{n+m}}{F_{n}}}=\varphi ^{m}} , because the ratios between consecutive Fibonacci numbers approaches φ {\displaystyle \varphi } . Since

2592-553: The sequence begins 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, also 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

2646-410: The start of the 19th century to 15,000 in 1900. At the beginning of the 19th century, the city district Königsborn (then Bad Königsborn ) gained prominence as a health resort with mineral springs . The cityscape of Königsborn still shows many historic buildings from that era, and the former spa gardens still serve as a recreation place for locals and tourists. In 2013, a geological survey showed that

2700-521: The starting values U 0 {\displaystyle U_{0}} and U 1 {\displaystyle U_{1}} , unless U 1 = − U 0 / φ {\displaystyle U_{1}=-U_{0}/\varphi } . This can be verified using Binet's formula . For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ... . The ratio of consecutive terms in this sequence shows

2754-411: The value F 0 = 0 {\displaystyle F_{0}=0} is omitted, so that the sequence starts with F 1 = F 2 = 1 , {\displaystyle F_{1}=F_{2}=1,} and the recurrence F n = F n − 1 + F n − 2 {\displaystyle F_{n}=F_{n-1}+F_{n-2}}

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2808-402: The variation] ... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21] ... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations]. Hemachandra (c. 1150) is credited with knowledge of the sequence as well, writing that "the sum of the last and the one before

2862-657: Was already known by Abraham de Moivre and Daniel Bernoulli : F n = φ n − ψ n φ − ψ = φ n − ψ n 5 , {\displaystyle F_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\varphi -\psi }}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}},} where φ = 1 + 5 2 ≈ 1.61803 39887 … {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803\,39887\ldots }

2916-715: Was opened in 1855 as part of the Dortmund–Soest railway built by the Bergisch-Märkische Railway Company (BME) and equipped with an impressive station building, which was sold for non-rail purposes in 2005. In 1866, the BME opened the line from Unna to Hamm to connect with the Cologne-Minden trunk line . Later the line was extended from Unna to Hagen, making Unna station into a railway junction of regional importance. Between 1899 and 1901

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