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| In [[elementary number theory]], a '''centered square number''' is a [[centered polygonal number|centered]] [[figurate number]] that gives the number of dots in a [[Square (geometry)|square]] with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given [[taxicab geometry|city block distance]] of the center dot on a regular [[square lattice]]. While centered square numbers, like [[figurate number]]s in general, have few if any direct practical applications, they are sometimes studied in [[recreational mathematics]] for their elegant geometric and arithmetic properties.
| | <br><br>I do not ordinarily look at myself. I love sharing interesting articles with other folks. I also like to examine the world wide web mainly because I think you will find a lot to find out in existence. When I am around the world-wide-web.<br>|<br>Name: Lilla Vincent<br>My age: 30 years old<br>Country: Germany<br>Home town: Kirchenlamitz <br>Post code: 95154<br>Address: Billwerder Neuer Deich 61<br>|<br>My name is Lilla Vincent. I life in Kirchenlamitz (Germany).<br>|<br>I’m Lilla from Kirchenlamitz studying American Politics. I did my schooling, secured 71% and hope to find someone with same interests in Leaf collecting and pressing.<br>|<br>I like my hobby Leaf collecting and pressing. Sounds boring? Not at all!<br>I try to learn French in my spare time. <br>|<br>My name is Lilla (41 years old) and my hobbies are Cooking and Board sports.<br>|<br>Hello, dear friend! I am Lilla. I smile that I could join to the entire world. I live in Germany, in the south region. I dream to see the various nations, to obtain acquainted with appealing individuals.<br>|<br>I'm Lilla (30) from Kirchenlamitz, Germany. <br>I'm learning French literature at a local university and I'm just about to graduate.<br>I have a part time job in a post office.<br>|<br>Hi! <br>My name is Lilla and I'm a 23 years old girl from Germany.<br>|<br>Hello! My name is Lilla. <br>It is a little about myself: I live in Germany, my city of Kirchenlamitz. <br>It's called often Northern or cultural capital of BY. I've married 1 years ago.<br>I have 2 children - a son (Elliott) and the daughter (Evelyne). We all like Leaf collecting and pressing.<br>|<br>Hi, everybody! <br>I'm French male :D. <br>I really like NCIS!<br>|<br>Hello from Germany. I'm glad to came here. My first name is Lilla. <br>I live in a small town called Kirchenlamitz in east Germany.<br>I was also born in Kirchenlamitz 33 years ago. Married in September year 2001. I'm working at the post office.<br>|<br>My name's Lilla Vincent but everybody calls me Lilla. I'm from Germany. I'm studying at the college (final year) and I play the Tuba for 7 years. Usually I choose songs from the famous films ;). <br>I have two brothers. I like Woodworking, watching movies and Auto racing.<br>|<br>I'm a 38 years old and study at the high school (American Politics).<br>In my free time I try to teach myself French. I've been twicethere and look forward to go there sometime in the future. I love to read, preferably on my kindle. I like to watch The Simpsons and Grey's Anatomy as well as docus about anything geological. I enjoy Leaf collecting and pressing.<br>|<br>I'm Lilla and I live in Kirchenlamitz. <br>I'm interested in American Politics, Leaf collecting and pressing and French art. I like to travel and reading fantasy.<br>|<br>I'm Lilla and I live with my husband and our two children in Kirchenlamitz, in the BY south part. My hobbies are Genealogy, Backpacking and Mountain biking.<br>|<br>Hello, I'm Lilla, a 23 year old from Kirchenlamitz, Germany.<br>My hobbies include (but are not limited to) Element collecting, Squash and watching NCIS.<br>|<br>Hi there! :) My name is Lilla, I'm a student studying American Politics from Kirchenlamitz, Germany.<br>|<br>I'm Lilla and I live in a seaside city in northern Germany, Kirchenlamitz. I'm 32 and I'm will soon finish my study at American Politics.<br>|<br>I am Lilla from Kirchenlamitz. I am learning to play the Tuba. Other hobbies are Leaf collecting and pressing.<br>|<br>I am Lilla and was born on 20 September 1975. My hobbies are Bird watching and Mountain biking.<br>|<br>My name is Lilla and I am studying Japanese Studies and Creative Writing at Kirchenlamitz / Germany. <br>}<br><br>my weblog ... laptop repair in schaumburg; [http://www.fotka.pl/link.php?u=http://www.computerrepairinschaumburg.us www.fotka.pl], |
| | |
| The figures for the first four centered square numbers are shown below:
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| {|
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| |- align="center" valign="middle"
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| |[[Image:GrayDot.svg|16px]]
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| |[[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]]
| |
| |
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| |[[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]]
| |
| |
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| |[[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]]
| |
| |- align="center" valign="top"
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| | <math>C_{4,1} = 1</math> | |
| |
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| | <math>C_{4,2} = 5</math>
| |
| |
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| | <math>C_{4,3} = 13</math> | |
| |
| |
| | <math>C_{4,4} = 25</math> | |
| |}
| |
| | |
| == Relationships with other figurate numbers ==
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| The ''n''th centered square number is given by the formula
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| :<math>C_{4,n} = n^2 + (n - 1)^2.\,</math>
| |
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| In other words, a centered square number is the sum of two consecutive [[square number]]s. The following pattern demonstrates this formula:
| |
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| {|
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| |- align="center" valign="middle"
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| |[[Image:GrayDot.svg|16px]]
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| |[[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
| |
| |
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| |[[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]]
| |
| |
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| |[[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
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| |- align="center" valign="top"
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| | <math>C_{4,1} = 0 + 1</math>
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| | <math>C_{4,2} = 1 + 4</math> | |
| |
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| | <math>C_{4,3} = 4 + 9</math>
| |
| |
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| | <math>C_{4,4} = 9 + 16</math>
| |
| |}
| |
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| The formula can also be expressed as
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| :<math>C_{4,n} = {(2n-1)^2 + 1 \over 2};</math> | |
| | |
| that is, ''n'' th centered square number is half of ''n'' th odd square number plus one, as illustrated below:
| |
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| {|
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| |- align="center" valign="bottom"
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| |[[Image:GrayDot.svg|16px]]
| |
| |
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| |[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]
| |
| |
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| |[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]
| |
| |
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| |[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]<br>[[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]][[Image:MissingDot.svg|16px]]
| |
| |- align="center" valign="top"
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| | <math>C_{4,1} = (1 + 1) / 2</math>
| |
| |
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| | <math>C_{4,2} = (9 + 1) / 2</math> | |
| |
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| | <math>C_{4,3} = (25 + 1) / 2</math>
| |
| |
| |
| | <math>C_{4,4} = (49 + 1) / 2</math> | |
| |}
| |
| | |
| Like all [[centered polygonal number]]s, centered square numbers can also be expressed in terms of [[triangular number]]s:
| |
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| :<math>C_{4,n} = 1 + 4\, T_{n-1},\,</math>
| |
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| where
| |
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| :<math>T_n = {n(n + 1) \over 2} = {n^2 + n \over 2} = {n+1 \choose 2}</math>
| |
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| is the ''n''th triangular number. This can be easily seen by removing the center dot and dividing the rest of the figure into four triangles, as below:
| |
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| {|
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| |- align="center" valign="middle"
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| |[[Image:BlackDot.svg|16px]]
| |
| |
| |
| |[[Image:RedDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:BlackDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
| |
| |
| |
| |[[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:BlackDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
| |
| |
| |
| |[[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:BlackDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:RedDot.svg|16px]][[Image:RedDot.svg|16px]]<br>[[Image:RedDot.svg|16px]] | |
| |- align="center" valign="top"
| |
| | <math>C_{4,1} = 1</math>
| |
| |
| |
| | <math>C_{4,2} = 1 + 4 \times 1</math>
| |
| |
| |
| | <math>C_{4,3} = 1 + 4 \times 3</math>
| |
| |
| |
| | <math>C_{4,4} = 1 + 4 \times 6.</math>
| |
| |}
| |
| | |
| The difference between two consecutive [[octahedral number]]s is a centered square number (Conway and Guy, p.50).
| |
| | |
| == Properties ==
| |
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| The first few centered square numbers are:
| |
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| :[[1 (number)|1]], [[5 (number)|5]], [[13 (number)|13]], [[25 (number)|25]], [[41 (number)|41]], [[61 (number)|61]], [[85 (number)|85]], [[113 (number)|113]], [[145 (number)|145]], 181, 221, 265, [[313 (number)|313]], [[365 (number)|365]], 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, … {{OEIS|id=A001844}}.
| |
| | |
| All centered square numbers are odd, and in base 10 one can notice the one's digits follows the pattern 1-5-3-5-1.
| |
| | |
| All centered square numbers and their divisors have a remainder of one when divided by four. Hence all centered square numbers and their divisors end with digits 1 or 5 in base [[senary|6]], [[octal|8]] or [[duodecimal|12]].
| |
| | |
| Every centered square number except 1 is the square of the third term of a leg–hypotenuse [[Pythagorean triple]] (for example, 3-4-5, 5-12-13).
| |
| | |
| === Centered square prime ===
| |
| | |
| A '''centered square prime''' is a centered square number that is [[prime number|prime]]. Unlike regular [[square number]]s, which are never prime, quite a few of the centered square numbers are prime. The first few centered square primes are:
| |
| | |
| :5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, … {{OEIS|id=A027862}}. A striking example can be seen in the 10th century al-Antaakii magic square.
| |
| | |
| == References ==
| |
| *{{citation
| |
| | last = Alfred | first = U.
| |
| | mr = 1571197
| |
| | issue = 3
| |
| | journal = Mathematics Magazine
| |
| | pages = 155–164
| |
| | title = {{math|''n''}} and {{math|''n'' + 1}} consecutive integers with equal sums of squares
| |
| | jstor = 2688938
| |
| | volume = 35
| |
| | year = 1962}}.
| |
| *{{Apostol IANT}}.
| |
| *{{citation
| |
| | last = Beiler | first = A. H.
| |
| | location = New York
| |
| | page = 125
| |
| | publisher = Dover
| |
| | title = Recreations in the Theory of Numbers
| |
| | year = 1964}}.
| |
| *{{citation
| |
| | last1 = Conway | first1 = John H. | author1-link = John Horton Conway
| |
| | last2 = Guy | first2 = Richard K. | author2-link = Richard K. Guy
| |
| | mr = 1411676
| |
| | isbn = 0-387-97993-X
| |
| | location = New York
| |
| | pages = 41–42
| |
| | publisher = Copernicus
| |
| | title = The Book of Numbers
| |
| | year = 1996}}.
| |
| | |
| ==External links==
| |
| | |
| * [http://www.muljadi.org/Median.htm (''n''<sup>2</sup> + 1) / 2 as a special case of ''M''(''i'', ''j'') = (''i''<sup>2</sup> + ''j'') / 2]
| |
| | |
| {{Classes of natural numbers}}
| |
| | |
| [[Category:Figurate numbers]]
| |
| [[Category:Quadrilaterals]]
| |
I do not ordinarily look at myself. I love sharing interesting articles with other folks. I also like to examine the world wide web mainly because I think you will find a lot to find out in existence. When I am around the world-wide-web.
|
Name: Lilla Vincent
My age: 30 years old
Country: Germany
Home town: Kirchenlamitz
Post code: 95154
Address: Billwerder Neuer Deich 61
|
My name is Lilla Vincent. I life in Kirchenlamitz (Germany).
|
I’m Lilla from Kirchenlamitz studying American Politics. I did my schooling, secured 71% and hope to find someone with same interests in Leaf collecting and pressing.
|
I like my hobby Leaf collecting and pressing. Sounds boring? Not at all!
I try to learn French in my spare time.
|
My name is Lilla (41 years old) and my hobbies are Cooking and Board sports.
|
Hello, dear friend! I am Lilla. I smile that I could join to the entire world. I live in Germany, in the south region. I dream to see the various nations, to obtain acquainted with appealing individuals.
|
I'm Lilla (30) from Kirchenlamitz, Germany.
I'm learning French literature at a local university and I'm just about to graduate.
I have a part time job in a post office.
|
Hi!
My name is Lilla and I'm a 23 years old girl from Germany.
|
Hello! My name is Lilla.
It is a little about myself: I live in Germany, my city of Kirchenlamitz.
It's called often Northern or cultural capital of BY. I've married 1 years ago.
I have 2 children - a son (Elliott) and the daughter (Evelyne). We all like Leaf collecting and pressing.
|
Hi, everybody!
I'm French male :D.
I really like NCIS!
|
Hello from Germany. I'm glad to came here. My first name is Lilla.
I live in a small town called Kirchenlamitz in east Germany.
I was also born in Kirchenlamitz 33 years ago. Married in September year 2001. I'm working at the post office.
|
My name's Lilla Vincent but everybody calls me Lilla. I'm from Germany. I'm studying at the college (final year) and I play the Tuba for 7 years. Usually I choose songs from the famous films ;).
I have two brothers. I like Woodworking, watching movies and Auto racing.
|
I'm a 38 years old and study at the high school (American Politics).
In my free time I try to teach myself French. I've been twicethere and look forward to go there sometime in the future. I love to read, preferably on my kindle. I like to watch The Simpsons and Grey's Anatomy as well as docus about anything geological. I enjoy Leaf collecting and pressing.
|
I'm Lilla and I live in Kirchenlamitz.
I'm interested in American Politics, Leaf collecting and pressing and French art. I like to travel and reading fantasy.
|
I'm Lilla and I live with my husband and our two children in Kirchenlamitz, in the BY south part. My hobbies are Genealogy, Backpacking and Mountain biking.
|
Hello, I'm Lilla, a 23 year old from Kirchenlamitz, Germany.
My hobbies include (but are not limited to) Element collecting, Squash and watching NCIS.
|
Hi there! :) My name is Lilla, I'm a student studying American Politics from Kirchenlamitz, Germany.
|
I'm Lilla and I live in a seaside city in northern Germany, Kirchenlamitz. I'm 32 and I'm will soon finish my study at American Politics.
|
I am Lilla from Kirchenlamitz. I am learning to play the Tuba. Other hobbies are Leaf collecting and pressing.
|
I am Lilla and was born on 20 September 1975. My hobbies are Bird watching and Mountain biking.
|
My name is Lilla and I am studying Japanese Studies and Creative Writing at Kirchenlamitz / Germany.
}
my weblog ... laptop repair in schaumburg; www.fotka.pl,