2197:
2506:
1641:
1475:
966:
Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones. Periodic zero and one sequences can be expressed as sums of trigonometric functions:
1281:
1123:
955:
806:
593:
355:
1506:
1845:
1289:
433:
300:
1718:
1131:
973:
632:
1498:
820:
2388:
680:
2031:
2378:
2471:
1674:
if it can be made periodic by dropping some finite number of terms from the beginning. Equivalently, the last condition can be stated as
1636:{\displaystyle \sum _{k=1}^{N}\cos \left(2\pi {\frac {n(k-1)}{N}}\right)/N=0,0,0,\cdots ,1,\cdots \quad {\text{sequence with period }}N}
484:
315:
2312:
1788:
2322:
2486:
2317:
2077:
2024:
2466:
2476:
1882:
2368:
2358:
1470:{\displaystyle \sum _{k=1}^{3}\cos \left(2\pi {\frac {n(k-1)}{3}}\right)/3=0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,\cdots }
1854:
1 / 3, 2 / 3, 1 / 4, 3 / 4, 1 / 5, 4 / 5, ...
2530:
2481:
2383:
2017:
1892:
373:
2509:
1858:
is asymptotically periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, ....
2491:
1887:
2373:
252:
1732:
1 / 56 = 0 . 0 1 7 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 ...
2363:
2353:
2343:
1276:{\displaystyle \sum _{k=1}^{2}\cos \left(2\pi {\frac {n(k-1)}{2}}\right)/2=0,1,0,1,0,1,0,1,0,\cdots }
1118:{\displaystyle \sum _{k=1}^{1}\cos \left(-\pi {\frac {n(k-1)}{1}}\right)/1=1,1,1,1,1,1,1,1,1,\cdots }
1677:
2458:
2280:
2120:
2067:
1728:. For example, the sequence of digits in the decimal expansion of 1/56 is eventually periodic:
1647:
211:
601:
2327:
2072:
1483:
950:{\displaystyle \prod _{n=1}^{kp+m}a_{n}=({\prod _{n=1}^{p}a_{n}})^{k}*\prod _{n=1}^{m}a_{n}}
2438:
2275:
2044:
641:
443:
8:
2418:
2285:
447:
2348:
2259:
2244:
2216:
2196:
2135:
1969:
2448:
2249:
2175:
2165:
2145:
2130:
1988:
1973:
1961:
653:
219:
2433:
2254:
2180:
2170:
2150:
2052:
1951:
801:{\displaystyle \sum _{n=1}^{kp+m}a_{n}=k*\sum _{n=1}^{p}a_{n}+\sum _{n=1}^{m}a_{n}}
476:
1907:
2211:
2140:
661:
361:
2443:
2428:
2423:
2102:
2087:
1956:
1939:
454:
215:
2524:
2408:
2082:
1965:
1655:
1651:
439:
2413:
2155:
2097:
1740:
if its terms approach those of a periodic sequence. That is, the sequence
40:
1761:, ... is asymptotically periodic if there exists a periodic sequence
442:
is periodic. The same holds true for the powers of any element of finite
2160:
2107:
20:
1940:"Periodicity of solutions of nonhomogeneous linear difference equations"
2009:
657:
360:
More generally, the sequence of digits in the decimal expansion of any
2092:
128:
2040:
36:
588:{\displaystyle x,\,f(x),\,f(f(x)),\,f^{3}(x),\,f^{4}(x),\,\ldots }
306:
367:
The sequence of powers of −1 is periodic with period two:
350:{\displaystyle {\frac {1}{7}}=0.142857\,142857\,142857\,\ldots }
1646:
One standard approach for proving these identities is to apply
16:
Sequence for which the same terms are repeated over and over
1840:{\displaystyle \lim _{n\rightarrow \infty }x_{n}-a_{n}=0.}
218:, then a periodic sequence is simply a special type of
2389:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
2379:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
1791:
1680:
1509:
1486:
1292:
1134:
976:
823:
683:
604:
487:
376:
318:
255:
664:
is the algorithmic problem of finding such a point.
1937:
1839:
1712:
1654:. Such sequences are foundational in the study of
1635:
1492:
1469:
1275:
1117:
949:
800:
626:
587:
427:
349:
294:
652:. Periodic points are important in the theory of
2522:
1793:
1938:Janglajew, Klara; Schmeidel, Ewa (2012-11-14).
438:More generally, the sequence of powers of any
2025:
2472:Hypergeometric function of a matrix argument
309:expansion of 1/7 is periodic with period 6:
2328:1 + 1/2 + 1/3 + ... (Riemann zeta function)
2032:
2018:
961:
2384:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
1955:
581:
558:
535:
510:
494:
343:
339:
335:
2039:
957:Where k and m<p are natural numbers.
808:Where k and m<p are natural numbers.
246:Every constant function is 1-periodic.
2523:
428:{\displaystyle -1,1,-1,1,-1,1,\ldots }
2013:
1986:
1933:
1931:
1877:
1875:
1873:
1871:
364:is eventually periodic (see below).
2349:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
812:
13:
1908:"Complexity of Periodic Sequences"
1803:
1661:
14:
2542:
2467:Generalized hypergeometric series
1987:Hosch, William L. (1 June 2018).
1928:
1905:
1868:
302:is periodic with least period 2.
295:{\displaystyle 1,2,1,2,1,2\dots }
226:for which a periodic sequence is
210:. If a sequence is regarded as a
2505:
2504:
2477:Lauricella hypergeometric series
2195:
1944:Advances in Difference Equations
660:to itself has a periodic point;
123:of repeated terms is called the
2487:Riemann's differential equation
1624:
672:
598:is a periodic sequence. Here,
1980:
1899:
1883:"Ultimately periodic sequence"
1800:
1566:
1554:
1349:
1337:
1191:
1179:
1033:
1021:
904:
867:
621:
615:
575:
569:
552:
546:
529:
526:
520:
514:
504:
498:
305:The sequence of digits in the
1:
2482:Modular hypergeometric series
2323:1/4 + 1/16 + 1/64 + 1/256 + ⋯
1861:
1713:{\displaystyle a_{k+r}=a_{k}}
667:
134:
43:are repeated over and over:
7:
2492:Theta hypergeometric series
1888:Encyclopedia of Mathematics
241:
214:whose domain is the set of
10:
2547:
2374:Infinite arithmetic series
2318:1/2 + 1/4 + 1/8 + 1/16 + ⋯
2313:1/2 − 1/4 + 1/8 − 1/16 + ⋯
1957:10.1186/1687-1847-2012-195
1850:For example, the sequence
1627:sequence with period
2500:
2457:
2401:
2336:
2305:
2298:
2268:
2237:
2230:
2204:
2193:
2116:
2060:
2051:
656:. Every function from a
627:{\displaystyle f^{n}(x)}
230:-periodic is called its
2205:Properties of sequences
1993:Encyclopedia Britannica
1738:asymptotically periodic
1724:and sufficiently large
1493:{\displaystyle \cdots }
962:Periodic 0, 1 sequences
2068:Arithmetic progression
1841:
1714:
1637:
1530:
1494:
1471:
1313:
1277:
1155:
1119:
997:
951:
936:
891:
853:
802:
787:
753:
713:
628:
589:
429:
351:
296:
2459:Hypergeometric series
2073:Geometric progression
1842:
1782:, ... for which
1715:
1650:to the corresponding
1638:
1510:
1495:
1472:
1293:
1278:
1135:
1120:
977:
952:
916:
871:
824:
803:
767:
733:
684:
629:
590:
430:
352:
297:
2531:Sequences and series
2439:Trigonometric series
2231:Properties of series
2078:Harmonic progression
1789:
1678:
1507:
1484:
1290:
1132:
974:
821:
681:
602:
485:
374:
316:
253:
27:(sometimes called a
2419:Formal power series
1672:ultimately periodic
1668:eventually periodic
1648:De Moivre's formula
39:for which the same
2217:Monotonic function
2136:Fibonacci sequence
1837:
1807:
1710:
1633:
1490:
1467:
1273:
1115:
947:
798:
624:
585:
425:
347:
292:
206:for all values of
2518:
2517:
2449:Generating series
2397:
2396:
2369:1 − 2 + 4 − 8 + ⋯
2364:1 + 2 + 4 + 8 + ⋯
2359:1 − 2 + 3 − 4 + ⋯
2354:1 + 2 + 3 + 4 + ⋯
2344:1 + 1 + 1 + 1 + ⋯
2294:
2293:
2222:Periodic sequence
2191:
2190:
2176:Triangular number
2166:Pentagonal number
2146:Heptagonal number
2131:Complete sequence
2053:Integer sequences
1989:"Rational number"
1792:
1628:
1573:
1356:
1198:
1040:
654:dynamical systems
327:
220:periodic function
178:, ... satisfying
155:periodic sequence
141:(purely) periodic
25:periodic sequence
2538:
2508:
2507:
2434:Dirichlet series
2303:
2302:
2235:
2234:
2199:
2171:Polygonal number
2151:Hexagonal number
2124:
2058:
2057:
2034:
2027:
2020:
2011:
2010:
2004:
2003:
2001:
1999:
1984:
1978:
1977:
1959:
1935:
1926:
1925:
1923:
1921:
1912:
1903:
1897:
1896:
1879:
1846:
1844:
1843:
1838:
1830:
1829:
1817:
1816:
1806:
1719:
1717:
1716:
1711:
1709:
1708:
1696:
1695:
1642:
1640:
1639:
1634:
1629:
1626:
1584:
1579:
1575:
1574:
1569:
1549:
1529:
1524:
1499:
1497:
1496:
1491:
1476:
1474:
1473:
1468:
1367:
1362:
1358:
1357:
1352:
1332:
1312:
1307:
1282:
1280:
1279:
1274:
1209:
1204:
1200:
1199:
1194:
1174:
1154:
1149:
1124:
1122:
1121:
1116:
1051:
1046:
1042:
1041:
1036:
1016:
996:
991:
956:
954:
953:
948:
946:
945:
935:
930:
912:
911:
902:
901:
900:
890:
885:
863:
862:
852:
838:
813:Partial Products
807:
805:
804:
799:
797:
796:
786:
781:
763:
762:
752:
747:
723:
722:
712:
698:
651:
647:
640:
638:
633:
631:
630:
625:
614:
613:
594:
592:
591:
586:
568:
567:
545:
544:
474:
470:
434:
432:
431:
426:
356:
354:
353:
348:
328:
320:
301:
299:
298:
293:
157:, is a sequence
2546:
2545:
2541:
2540:
2539:
2537:
2536:
2535:
2521:
2520:
2519:
2514:
2496:
2453:
2402:Kinds of series
2393:
2332:
2299:Explicit series
2290:
2264:
2226:
2212:Cauchy sequence
2200:
2187:
2141:Figurate number
2118:
2112:
2103:Powers of three
2047:
2038:
2008:
2007:
1997:
1995:
1985:
1981:
1936:
1929:
1919:
1917:
1910:
1904:
1900:
1881:
1880:
1869:
1864:
1825:
1821:
1812:
1808:
1796:
1790:
1787:
1786:
1781:
1774:
1767:
1760:
1753:
1746:
1704:
1700:
1685:
1681:
1679:
1676:
1675:
1664:
1662:Generalizations
1625:
1580:
1550:
1548:
1541:
1537:
1525:
1514:
1508:
1505:
1504:
1485:
1482:
1481:
1363:
1333:
1331:
1324:
1320:
1308:
1297:
1291:
1288:
1287:
1205:
1175:
1173:
1166:
1162:
1150:
1139:
1133:
1130:
1129:
1047:
1017:
1015:
1008:
1004:
992:
981:
975:
972:
971:
964:
941:
937:
931:
920:
907:
903:
896:
892:
886:
875:
870:
858:
854:
839:
828:
822:
819:
818:
815:
792:
788:
782:
771:
758:
754:
748:
737:
718:
714:
699:
688:
682:
679:
678:
675:
670:
662:cycle detection
649:
645:
636:
635:
609:
605:
603:
600:
599:
563:
559:
540:
536:
486:
483:
482:
472:
458:
457:for a function
375:
372:
371:
362:rational number
319:
317:
314:
313:
254:
251:
250:
244:
222:. The smallest
216:natural numbers
202:
193:
177:
170:
163:
143:sequence (with
137:
114:
105:
98:
91:
82:
75:
68:
59:
52:
17:
12:
11:
5:
2544:
2534:
2533:
2516:
2515:
2513:
2512:
2501:
2498:
2497:
2495:
2494:
2489:
2484:
2479:
2474:
2469:
2463:
2461:
2455:
2454:
2452:
2451:
2446:
2444:Fourier series
2441:
2436:
2431:
2429:Puiseux series
2426:
2424:Laurent series
2421:
2416:
2411:
2405:
2403:
2399:
2398:
2395:
2394:
2392:
2391:
2386:
2381:
2376:
2371:
2366:
2361:
2356:
2351:
2346:
2340:
2338:
2334:
2333:
2331:
2330:
2325:
2320:
2315:
2309:
2307:
2300:
2296:
2295:
2292:
2291:
2289:
2288:
2283:
2278:
2272:
2270:
2266:
2265:
2263:
2262:
2257:
2252:
2247:
2241:
2239:
2232:
2228:
2227:
2225:
2224:
2219:
2214:
2208:
2206:
2202:
2201:
2194:
2192:
2189:
2188:
2186:
2185:
2184:
2183:
2173:
2168:
2163:
2158:
2153:
2148:
2143:
2138:
2133:
2127:
2125:
2114:
2113:
2111:
2110:
2105:
2100:
2095:
2090:
2085:
2080:
2075:
2070:
2064:
2062:
2055:
2049:
2048:
2037:
2036:
2029:
2022:
2014:
2006:
2005:
1979:
1927:
1915:www.math.ru.nl
1898:
1866:
1865:
1863:
1860:
1856:
1855:
1848:
1847:
1836:
1833:
1828:
1824:
1820:
1815:
1811:
1805:
1802:
1799:
1795:
1779:
1772:
1765:
1758:
1751:
1744:
1736:A sequence is
1734:
1733:
1707:
1703:
1699:
1694:
1691:
1688:
1684:
1666:A sequence is
1663:
1660:
1644:
1643:
1632:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1583:
1578:
1572:
1568:
1565:
1562:
1559:
1556:
1553:
1547:
1544:
1540:
1536:
1533:
1528:
1523:
1520:
1517:
1513:
1501:
1500:
1489:
1478:
1477:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1366:
1361:
1355:
1351:
1348:
1345:
1342:
1339:
1336:
1330:
1327:
1323:
1319:
1316:
1311:
1306:
1303:
1300:
1296:
1284:
1283:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1208:
1203:
1197:
1193:
1190:
1187:
1184:
1181:
1178:
1172:
1169:
1165:
1161:
1158:
1153:
1148:
1145:
1142:
1138:
1126:
1125:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1050:
1045:
1039:
1035:
1032:
1029:
1026:
1023:
1020:
1014:
1011:
1007:
1003:
1000:
995:
990:
987:
984:
980:
963:
960:
959:
958:
944:
940:
934:
929:
926:
923:
919:
915:
910:
906:
899:
895:
889:
884:
881:
878:
874:
869:
866:
861:
857:
851:
848:
845:
842:
837:
834:
831:
827:
814:
811:
810:
809:
795:
791:
785:
780:
777:
774:
770:
766:
761:
757:
751:
746:
743:
740:
736:
732:
729:
726:
721:
717:
711:
708:
705:
702:
697:
694:
691:
687:
674:
671:
669:
666:
623:
620:
617:
612:
608:
596:
595:
584:
580:
577:
574:
571:
566:
562:
557:
554:
551:
548:
543:
539:
534:
531:
528:
525:
522:
519:
516:
513:
509:
506:
503:
500:
497:
493:
490:
455:periodic point
436:
435:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
358:
357:
346:
342:
338:
334:
331:
326:
323:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
261:
258:
243:
240:
204:
203:
198:
185:
175:
168:
161:
136:
133:
117:
116:
110:
103:
96:
87:
80:
73:
64:
57:
50:
15:
9:
6:
4:
3:
2:
2543:
2532:
2529:
2528:
2526:
2511:
2503:
2502:
2499:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2473:
2470:
2468:
2465:
2464:
2462:
2460:
2456:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2409:Taylor series
2407:
2406:
2404:
2400:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2341:
2339:
2335:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2310:
2308:
2304:
2301:
2297:
2287:
2284:
2282:
2279:
2277:
2274:
2273:
2271:
2267:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2242:
2240:
2236:
2233:
2229:
2223:
2220:
2218:
2215:
2213:
2210:
2209:
2207:
2203:
2198:
2182:
2179:
2178:
2177:
2174:
2172:
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2128:
2126:
2122:
2115:
2109:
2106:
2104:
2101:
2099:
2098:Powers of two
2096:
2094:
2091:
2089:
2086:
2084:
2083:Square number
2081:
2079:
2076:
2074:
2071:
2069:
2066:
2065:
2063:
2059:
2056:
2054:
2050:
2046:
2042:
2035:
2030:
2028:
2023:
2021:
2016:
2015:
2012:
1994:
1990:
1983:
1975:
1971:
1967:
1963:
1958:
1953:
1949:
1945:
1941:
1934:
1932:
1916:
1909:
1906:Bosma, Wieb.
1902:
1894:
1890:
1889:
1884:
1878:
1876:
1874:
1872:
1867:
1859:
1853:
1852:
1851:
1834:
1831:
1826:
1822:
1818:
1813:
1809:
1797:
1785:
1784:
1783:
1778:
1771:
1764:
1757:
1750:
1743:
1739:
1731:
1730:
1729:
1727:
1723:
1705:
1701:
1697:
1692:
1689:
1686:
1682:
1673:
1669:
1659:
1657:
1656:number theory
1653:
1652:root of unity
1649:
1630:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1581:
1576:
1570:
1563:
1560:
1557:
1551:
1545:
1542:
1538:
1534:
1531:
1526:
1521:
1518:
1515:
1511:
1503:
1502:
1487:
1480:
1479:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1364:
1359:
1353:
1346:
1343:
1340:
1334:
1328:
1325:
1321:
1317:
1314:
1309:
1304:
1301:
1298:
1294:
1286:
1285:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1206:
1201:
1195:
1188:
1185:
1182:
1176:
1170:
1167:
1163:
1159:
1156:
1151:
1146:
1143:
1140:
1136:
1128:
1127:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1048:
1043:
1037:
1030:
1027:
1024:
1018:
1012:
1009:
1005:
1001:
998:
993:
988:
985:
982:
978:
970:
969:
968:
942:
938:
932:
927:
924:
921:
917:
913:
908:
897:
893:
887:
882:
879:
876:
872:
864:
859:
855:
849:
846:
843:
840:
835:
832:
829:
825:
817:
816:
793:
789:
783:
778:
775:
772:
768:
764:
759:
755:
749:
744:
741:
738:
734:
730:
727:
724:
719:
715:
709:
706:
703:
700:
695:
692:
689:
685:
677:
676:
665:
663:
659:
655:
643:
618:
610:
606:
582:
578:
572:
564:
560:
555:
549:
541:
537:
532:
523:
517:
511:
507:
501:
495:
491:
488:
481:
480:
479:
478:
469:
465:
461:
456:
451:
449:
445:
441:
440:root of unity
422:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
377:
370:
369:
368:
365:
363:
344:
340:
336:
332:
329:
324:
321:
312:
311:
310:
308:
303:
289:
286:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
249:The sequence
247:
239:
237:
233:
229:
225:
221:
217:
213:
209:
201:
197:
192:
188:
184:
181:
180:
179:
174:
167:
160:
156:
154:
149:
148:
142:
132:
130:
126:
122:
113:
109:
102:
95:
92:,
90:
86:
79:
72:
69:,
67:
63:
56:
49:
46:
45:
44:
42:
38:
34:
30:
26:
22:
2414:Power series
2221:
2156:Lucas number
2108:Powers of 10
2088:Cubic number
1996:. Retrieved
1992:
1982:
1947:
1943:
1918:. Retrieved
1914:
1901:
1886:
1857:
1849:
1776:
1769:
1762:
1755:
1748:
1741:
1737:
1735:
1725:
1721:
1671:
1667:
1665:
1645:
965:
673:Partial Sums
597:
467:
463:
459:
452:
437:
366:
359:
304:
248:
245:
236:exact period
235:
232:least period
231:
227:
223:
207:
205:
199:
195:
190:
186:
182:
172:
165:
158:
152:
151:
146:
144:
140:
138:
124:
120:
118:
111:
107:
100:
93:
88:
84:
77:
70:
65:
61:
54:
47:
32:
28:
24:
18:
2281:Conditional
2269:Convergence
2260:Telescoping
2245:Alternating
2161:Pell number
648:applied to
642:composition
471:is a point
119:The number
21:mathematics
2306:Convergent
2250:Convergent
1950:(1): 195.
1862:References
668:Identities
658:finite set
634:means the
135:Definition
2337:Divergent
2255:Divergent
2117:Advanced
2093:Factorial
2041:Sequences
1998:13 August
1974:122892501
1966:1687-1847
1920:13 August
1893:EMS Press
1819:−
1804:∞
1801:→
1720:for some
1622:⋯
1610:⋯
1561:−
1546:π
1535:
1512:∑
1488:⋯
1465:⋯
1344:−
1329:π
1318:
1295:∑
1271:⋯
1186:−
1171:π
1160:
1137:∑
1113:⋯
1028:−
1013:π
1010:−
1002:
979:∑
918:∏
914:∗
873:∏
826:∏
769:∑
735:∑
731:∗
686:∑
583:…
423:…
408:−
393:−
378:−
345:…
290:…
2525:Category
2510:Category
2276:Absolute
462: :
333:0.142857
242:Examples
212:function
150:), or a
37:sequence
2286:Uniform
1895:, 2001
1775:,
1768:,
1754:,
1747:,
307:decimal
145:period
106:, ...,
83:, ...,
60:, ...,
35:) is a
2238:Series
2045:series
1972:
1964:
475:whose
341:142857
337:142857
129:period
125:period
2181:array
2061:Basic
1970:S2CID
1911:(PDF)
639:-fold
477:orbit
448:group
446:in a
444:order
115:, ...
41:terms
33:orbit
29:cycle
2121:list
2043:and
2000:2021
1962:ISSN
1948:2012
1922:2021
23:, a
1952:doi
1794:lim
1670:or
1532:cos
1315:cos
1157:cos
999:cos
644:of
234:or
131:).
31:or
19:In
2527::
1991:.
1968:.
1960:.
1946:.
1942:.
1930:^
1913:.
1891:,
1885:,
1870:^
1835:0.
1658:.
466:→
453:A
450:.
238:.
194:=
171:,
164:,
153:p-
139:A
99:,
76:,
53:,
2123:)
2119:(
2033:e
2026:t
2019:v
2002:.
1976:.
1954::
1924:.
1832:=
1827:n
1823:a
1814:n
1810:x
1798:n
1780:3
1777:a
1773:2
1770:a
1766:1
1763:a
1759:3
1756:x
1752:2
1749:x
1745:1
1742:x
1726:k
1722:r
1706:k
1702:a
1698:=
1693:r
1690:+
1687:k
1683:a
1631:N
1619:,
1616:1
1613:,
1607:,
1604:0
1601:,
1598:0
1595:,
1592:0
1589:=
1586:N
1582:/
1577:)
1571:N
1567:)
1564:1
1558:k
1555:(
1552:n
1543:2
1539:(
1527:N
1522:1
1519:=
1516:k
1462:,
1459:1
1456:,
1453:0
1450:,
1447:0
1444:,
1441:1
1438:,
1435:0
1432:,
1429:0
1426:,
1423:1
1420:,
1417:0
1414:,
1411:0
1408:,
1405:1
1402:,
1399:0
1396:,
1393:0
1390:,
1387:1
1384:,
1381:0
1378:,
1375:0
1372:=
1369:3
1365:/
1360:)
1354:3
1350:)
1347:1
1341:k
1338:(
1335:n
1326:2
1322:(
1310:3
1305:1
1302:=
1299:k
1268:,
1265:0
1262:,
1259:1
1256:,
1253:0
1250:,
1247:1
1244:,
1241:0
1238:,
1235:1
1232:,
1229:0
1226:,
1223:1
1220:,
1217:0
1214:=
1211:2
1207:/
1202:)
1196:2
1192:)
1189:1
1183:k
1180:(
1177:n
1168:2
1164:(
1152:2
1147:1
1144:=
1141:k
1110:,
1107:1
1104:,
1101:1
1098:,
1095:1
1092:,
1089:1
1086:,
1083:1
1080:,
1077:1
1074:,
1071:1
1068:,
1065:1
1062:,
1059:1
1056:=
1053:1
1049:/
1044:)
1038:1
1034:)
1031:1
1025:k
1022:(
1019:n
1006:(
994:1
989:1
986:=
983:k
943:n
939:a
933:m
928:1
925:=
922:n
909:k
905:)
898:n
894:a
888:p
883:1
880:=
877:n
868:(
865:=
860:n
856:a
850:m
847:+
844:p
841:k
836:1
833:=
830:n
794:n
790:a
784:m
779:1
776:=
773:n
765:+
760:n
756:a
750:p
745:1
742:=
739:n
728:k
725:=
720:n
716:a
710:m
707:+
704:p
701:k
696:1
693:=
690:n
650:x
646:f
637:n
622:)
619:x
616:(
611:n
607:f
579:,
576:)
573:x
570:(
565:4
561:f
556:,
553:)
550:x
547:(
542:3
538:f
533:,
530:)
527:)
524:x
521:(
518:f
515:(
512:f
508:,
505:)
502:x
499:(
496:f
492:,
489:x
473:x
468:X
464:X
460:f
420:,
417:1
414:,
411:1
405:,
402:1
399:,
396:1
390:,
387:1
384:,
381:1
330:=
325:7
322:1
287:2
284:,
281:1
278:,
275:2
272:,
269:1
266:,
263:2
260:,
257:1
228:p
224:p
208:n
200:n
196:a
191:p
189:+
187:n
183:a
176:3
173:a
169:2
166:a
162:1
159:a
147:p
127:(
121:p
112:p
108:a
104:2
101:a
97:1
94:a
89:p
85:a
81:2
78:a
74:1
71:a
66:p
62:a
58:2
55:a
51:1
48:a
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