"페리 수열"의 두 판 사이의 차이

2020년 1월 5일 (일) 23:11 기준 최신판

1 개요[ | ]

Farey sequence
페리 수열, 패리 수열

0, 1, 그 사이에 있는 분모가 어떤 자연수 n 을 넘지 않는 기약진분수를 오름차순으로 나열한 수열
[math]\displaystyle{ F_n: 0 \le h \le k \le n }[/math]이고 [math]\displaystyle{ \gcd(h,k)=1 }[/math]을 만족하는 [math]\displaystyle{ \dfrac{h}{k} }[/math]를 오름차순으로 나열한 수열

2 표[ | ]

n=1…8까지의 페리 수열

[math]\displaystyle{ F_1 = \left\{ \dfrac{0}{1}, \dfrac{1}{1} \right\} }[/math]

[math]\displaystyle{ F_2 = \left\{ \dfrac{0}{1}, \dfrac{1}{2}, \dfrac{1}{1} \right\} }[/math]

[math]\displaystyle{ F_3 = \left\{ \dfrac{0}{1}, \dfrac{1}{3}, \dfrac{1}{2}, \dfrac{2}{3}, \dfrac{1}{1} \right\} }[/math]

[math]\displaystyle{ F_4 = \left\{ \dfrac{0}{1}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{1}{1} \right\} }[/math]

[math]\displaystyle{ F_5 = \left\{ \dfrac{0}{1}, \dfrac{1}{5}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{2}{5}, \dfrac{1}{2}, \dfrac{3}{5}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{1}{1} \right\} }[/math]

[math]\displaystyle{ F_6 = \left\{ \dfrac{0}{1}, \dfrac{1}{6}, \dfrac{1}{5}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{2}{5}, \dfrac{1}{2}, \dfrac{3}{5}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{5}{6}, \dfrac{1}{1} \right\} }[/math]

[math]\displaystyle{ F_7 = \left\{ \dfrac{0}{1}, \dfrac{1}{7}, \dfrac{1}{6}, \dfrac{1}{5}, \dfrac{1}{4}, \dfrac{2}{7}, \dfrac{1}{3}, \dfrac{2}{5}, \dfrac{3}{7}, \dfrac{1}{2}, \dfrac{4}{7}, \dfrac{3}{5}, \dfrac{2}{3}, \dfrac{5}{7}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{5}{6}, \dfrac{6}{7}, \dfrac{1}{1} \right\} }[/math]

[math]\displaystyle{ F_8 = \left\{ \dfrac{0}{1}, \dfrac{1}{8}, \dfrac{1}{7}, \dfrac{1}{6}, \dfrac{1}{5}, \dfrac{1}{4}, \dfrac{2}{7}, \dfrac{1}{3}, \dfrac{3}{8}, \dfrac{2}{5}, \dfrac{3}{7}, \dfrac{1}{2}, \dfrac{4}{7}, \dfrac{3}{5}, \dfrac{5}{8}, \dfrac{2}{3}, \dfrac{5}{7}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{5}{6}, \dfrac{6}{7}, \dfrac{7}{8}, \dfrac{1}{1} \right\} }[/math]

3 같이 보기[ | ]

4 참고[ | ]

@@ 3번째 줄: / 3번째 줄: @@
 ;페리 수열, 패리 수열
 * 0, 1, 그 사이에 있는 분모가 어떤 자연수 n 을 넘지 않는 기약진분수를 오름차순으로 나열한 수열
-*<math>F_n:  0 \le h \le k \le n</math>이고 <math>\gcd(h,k)=1</math>을 만족하는 <math>\frac{h}{k}</math>를 오름차순으로 나열한 수열
+*<math>F_n:  0 \le h \le k \le n</math>이고 <math>\gcd(h,k)=1</math>을 만족하는 <math>\dfrac{h}{k}</math>를 오름차순으로 나열한 수열
 == 표 ==
 ''n''=1…8까지의 페리 수열
-:<math>F_1 = \left\{ \frac{0}{1}, \frac{1}{1} \right\}</math>
+:<math>F_1 = \left\{ \dfrac{0}{1}, \dfrac{1}{1} \right\}</math>
-:<math>F_2 = \left\{ \frac{0}{1}, \frac{1}{2}, \frac{1}{1} \right\}</math>
+:<math>F_2 = \left\{ \dfrac{0}{1}, \dfrac{1}{2}, \dfrac{1}{1} \right\}</math>
-:<math>F_3 = \left\{ \frac{0}{1}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{1}{1} \right\}</math>
+:<math>F_3 = \left\{ \dfrac{0}{1}, \dfrac{1}{3}, \dfrac{1}{2}, \dfrac{2}{3}, \dfrac{1}{1} \right\}</math>
-:<math>F_4 = \left\{ \frac{0}{1}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{1}{1} \right\}</math>
+:<math>F_4 = \left\{ \dfrac{0}{1}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{1}{1} \right\}</math>
-:<math>F_5 = \left\{ \frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{1}{1} \right\}</math>
+:<math>F_5 = \left\{ \dfrac{0}{1}, \dfrac{1}{5}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{2}{5}, \dfrac{1}{2}, \dfrac{3}{5}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{1}{1} \right\}</math>
-:<math>F_6 = \left\{ \frac{0}{1}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{1}{1} \right\}</math>
+:<math>F_6 = \left\{ \dfrac{0}{1}, \dfrac{1}{6}, \dfrac{1}{5}, \dfrac{1}{4}, \dfrac{1}{3}, \dfrac{2}{5}, \dfrac{1}{2}, \dfrac{3}{5}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{5}{6}, \dfrac{1}{1} \right\}</math>
-:<math>F_7 = \left\{ \frac{0}{1}, \frac{1}{7}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{2}{7}, \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{1}{2}, \frac{4}{7}, \frac{3}{5}, \frac{2}{3}, \frac{5}{7}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{1}{1} \right\}</math>
+:<math>F_7 = \left\{ \dfrac{0}{1}, \dfrac{1}{7}, \dfrac{1}{6}, \dfrac{1}{5}, \dfrac{1}{4}, \dfrac{2}{7}, \dfrac{1}{3}, \dfrac{2}{5}, \dfrac{3}{7}, \dfrac{1}{2}, \dfrac{4}{7}, \dfrac{3}{5}, \dfrac{2}{3}, \dfrac{5}{7}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{5}{6}, \dfrac{6}{7}, \dfrac{1}{1} \right\}</math>
-:<math>F_8 = \left\{ \frac{0}{1}, \frac{1}{8}, \frac{1}{7}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{2}{7}, \frac{1}{3}, \frac{3}{8}, \frac{2}{5}, \frac{3}{7}, \frac{1}{2}, \frac{4}{7}, \frac{3}{5}, \frac{5}{8}, \frac{2}{3}, \frac{5}{7}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}, \frac{1}{1} \right\}</math>
+:<math>F_8 = \left\{ \dfrac{0}{1}, \dfrac{1}{8}, \dfrac{1}{7}, \dfrac{1}{6}, \dfrac{1}{5}, \dfrac{1}{4}, \dfrac{2}{7}, \dfrac{1}{3}, \dfrac{3}{8}, \dfrac{2}{5}, \dfrac{3}{7}, \dfrac{1}{2}, \dfrac{4}{7}, \dfrac{3}{5}, \dfrac{5}{8}, \dfrac{2}{3}, \dfrac{5}{7}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{5}{6}, \dfrac{6}{7}, \dfrac{7}{8}, \dfrac{1}{1} \right\}</math>
 ==같이 보기==