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In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a 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 city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties. The figures for the first four centered square numbers are shown below: = 5 | | | | |} == Relationships with other figurate numbers == The ''n''th centered square number is given by the formula : In other words, a centered square number is the sum of two consecutive square numbers. The following pattern demonstrates this formula: = 1 + 4 | | | | |} The formula can also be expressed as : that is, ''n'' th centered square number is half of ''n'' th odd square number plus one, as illustrated below: = (9 + 1) / 2 | | | | |} Like all centered polygonal numbers, centered square numbers can also be expressed in terms of triangular numbers: : where : 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: = 1 + 4 \times 1 | | | | |} The difference between two consecutive octahedral numbers is a centered square number (Conway and Guy, p.50). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「centered square number」の詳細全文を読む スポンサード リンク
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