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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.
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| 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]]
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| |- 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>
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| | <math>C_{4,4} = 25</math>
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| |}
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| == 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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| |}
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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> | |
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| 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]]
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| |- 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>
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| | <math>C_{4,4} = (49 + 1) / 2</math>
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| |}
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| 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]]
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| |[[Image:RedDot.svg|16px]]<br>[[Image:GrayDot.svg|16px]][[Image:BlackDot.svg|16px]][[Image:GrayDot.svg|16px]]<br>[[Image:RedDot.svg|16px]]
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| |[[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]]
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| |[[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]]
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| |- align="center" valign="top"
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| | <math>C_{4,1} = 1</math>
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| | <math>C_{4,2} = 1 + 4 \times 1</math>
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| | <math>C_{4,3} = 1 + 4 \times 3</math>
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| | <math>C_{4,4} = 1 + 4 \times 6.</math>
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| |}
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| The difference between two consecutive [[octahedral number]]s is a centered square number (Conway and Guy, p.50).
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| == 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}}.
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| 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.
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| 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]].
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| 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).
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| === Centered square prime ===
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| 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:
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| :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.
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| == References ==
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| *{{citation
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| | last = Alfred | first = U.
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| | mr = 1571197
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| | issue = 3
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| | journal = Mathematics Magazine
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| | pages = 155–164
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| | title = {{math|''n''}} and {{math|''n'' + 1}} consecutive integers with equal sums of squares
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| | jstor = 2688938
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| | volume = 35
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| | year = 1962}}.
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| *{{Apostol IANT}}.
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| *{{citation
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| | last = Beiler | first = A. H.
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| | location = New York
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| | page = 125
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| | publisher = Dover
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| | title = Recreations in the Theory of Numbers
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| | year = 1964}}.
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| *{{citation
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| | last1 = Conway | first1 = John H. | author1-link = John Horton Conway
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| | last2 = Guy | first2 = Richard K. | author2-link = Richard K. Guy
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| | mr = 1411676
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| | isbn = 0-387-97993-X
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| | location = New York
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| | pages = 41–42
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| | publisher = Copernicus
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| | title = The Book of Numbers
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| | year = 1996}}.
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| ==External links==
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| * [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]
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| {{Classes of natural numbers}}
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| [[Category:Figurate numbers]]
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| [[Category:Quadrilaterals]]
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