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[讨论] 数列通项公式

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发表于 2021-4-5 18:38:28 | 显示全部楼层 |阅读模式

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$a_{n}=1-\frac{n}{a_{n-1}}$

大家帮忙找一个这样的数列

毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-4-5 23:38:19 | 显示全部楼层
  1. 23:35:12> a=[2,1]
  2. %7 = [2, 1]
  3. 23:35:16> a=f(a)
  4. %8 = [0, 2]
  5. 23:35:18> a=[3,1]
  6. %9 = [3, 1]
  7. 23:35:22> a=f(a)
  8. %10 = [1/3, 2]
  9. 23:35:23> a=f(a)
  10. %11 = [-8, 3]
  11. 23:35:24> a=f(a)
  12. %12 = [3/2, 4]
  13. 23:35:25> a=f(a)
  14. %13 = [-7/3, 5]
  15. 23:35:26> a=f(a)
  16. %14 = [25/7, 6]
  17. 23:35:27> a=f(a)
  18. %15 = [-24/25, 7]
  19. 23:35:28> a=f(a)
  20. %16 = [28/3, 8]
  21. 23:35:29> a=f(a)
  22. %17 = [1/28, 9]
  23. 23:35:29> a=f(a)
  24. %18 = [-279, 10]
  25. 23:35:30> a=f(a)
  26. %19 = [290/279, 11]
  27. 23:35:31> a=f(a)
  28. %20 = [-1529/145, 12]
  29. 23:35:31> a=f(a)
  30. %21 = [3414/1529, 13]
  31. 23:35:32> a=f(a)
  32. %22 = [-8996/1707, 14]
  33. 23:35:33> a=f(a)
  34. %23 = [34601/8996, 15]
  35. 23:35:33> a=f(a)
  36. %24 = [-109335/34601, 16]
  37. 23:35:34> a=f(a)
  38. %25 = [697552/109335, 17]
  39. 23:35:34> a=f(a)
  40. %26 = [-635239/348776, 18]
  41. 23:35:35> a=f(a)
  42. %27 = [7261983/635239, 19]
  43. 23:35:35> a=f(a)
  44. %28 = [-5442797/7261983, 20]
  45. 23:35:36> a=f(a)
  46. %29 = [157944440/5442797, 21]
  47. 23:35:36> a=f(a)
  48. %30 = [19101453/78972220, 22]
  49. 23:35:36> a=f(a)
  50. %31 = [-1797259607/19101453, 23]
  51. 23:35:37> a=f(a)
  52. %32 = [2255694479/1797259607, 24]
  53. 23:35:37> a=f(a)
  54. %33 = [-42675795696/2255694479, 25]
  55. 23:35:38>
  56. 23:35:38> a=f(a)
  57. %34 = [50661926075/21337897848, 26]
  58. 23:35:39> a=f(a)
  59. %35 = [-525461315821/50661926075, 27]
  60. 23:35:39> a=f(a)
  61. %36 = [1943995245921/525461315821, 28]
  62. 23:35:40> a=f(a)
  63. %37 = [-13294382912888/1943995245921, 29]
  64. 23:35:40> a=f(a)
  65. %38 = [35807120145259/6647191456444, 30]
  66. 23:35:41> a=f(a)
  67. %39 = [-170255815004505/35807120145259, 31]
  68. 23:35:41> a=f(a)
  69. %40 = [1316083659652793/170255815004505, 32]
  70. 23:35:42> a=f(a)
  71. %41 = [-4302358235495872/1316083659652793, 33]
  72. 23:35:42> a=f(a)
  73. %42 = [24524601331845417/2151179117747936, 34]
  74. 23:35:42> a=f(a)
  75. %43 = [-50766667789332343/24524601331845417, 35]
  76. 23:35:43>
  77. 23:37:13> a=f(a)
  78. %44 = [933652315735767355/50766667789332343, 36]
  79. 23:37:14> a=f(a)
  80. %45 = [-944714392469529336/933652315735767355, 37]
  81. 23:37:14> a=f(a)
  82. %46 = [18211751195214344413/472357196234764668, 38]
  83. 23:37:14> a=f(a)
  84. %47 = [-210179457941477639/18211751195214344413, 39]
  85. 23:37:15> a=f(a)
  86. %48 = [728680227266515254159/210179457941477639, 40]
  87. 23:37:17> a=f(a)
  88. %49 = [17562509011973528560/17772688469915006199, 41]
  89. 23:37:17> a=f(a)
  90. %50 = [-364445203362228365899/8781254505986764280, 42]
  91. 23:37:17> a=f(a)
  92. %51 = [742039147119659229939/364445203362228365899, 43]
  93. 23:37:17> a=f(a)
  94. %52 = [-15293549800818388869617/742039147119659229939, 44]
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我是找不到通项公式了……
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-4-6 16:03:55 | 显示全部楼层
这是有理分式。所以如果设初始值是$a_1=t$,那么 $a_n$一定是关于$t$的有理分式。不妨设$a_{n} = \frac{A_{n}+B_{n}t}{C_{n}+D_{n}t}$,于是得到递推公式$A_{n}=k*A_{n-1}-n*k^2*C_{n-1}, B_{n}=k*B_{n-1}-n*k^2*D_{n-1} ,C_{n}=k*A_{n-1}, D_{n} = k*B_{n-1}$
  1. {1,t}
  2. {2,1-2/t}
  3. {3,-2-6/(-2+t)}
  4. {4,3-6/(1+t)}
  5. {5,-(2/3)-10/(3 (-1+t))}
  6. {6,10-45/(4+t)}
  7. {7,3/10-63/(10 (-1+2 t))}
  8. {8,-(77/3)-280/(-11+t)}
  9. {9,104/77-3240/(77 (-1+11 t))}
  10. {10,-(333/52)-14175/(52 (-38+13 t))}
  11. {11,905/333-7700/(37 (13+37 t))}
  12. {12,-(3091/905)-299376/(181 (-311+181 t))}
  13. {13,14856/3091-8845200/(3091 (269+281 t))}
  14. {14,-(14209/7428)-4729725/(1238 (-719+619 t))}
  15. {15,125629/14209-392931000/(14209 (2257+1093 t))}
  16. {16,-(101715/125629)-1667952000/(17947 (-14297+17947 t))}
  17. {17,2237408/101715-6175128960/(6781 (37969+6781 t))}
  18. {18,203269/1118704-97692469875/(279676 (-35369+69919 t))}
  19. {19,-(21052107/203269)-635262264000/(203269 (-30437+1087 t))}
  20. {20,25117487/21052107-29331862560000/(2339123 (-562273+2339123 t))}
复制代码
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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