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数学研发论坛»论坛 › 【数学研究】 › 难题征解 › 疑似对数螺线的数列
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楼主: KeyTo9_Fans

[原创] 疑似对数螺线的数列

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mathe
发表于 2019-5-11 09:48:35 | 显示全部楼层
然后我们可以有精确通项公式
a(n)=-1+
  (0.13741867082740688265826177161980396845 - 0.27477457933438246604681642538828208234*I)*(-0.43338019958693104649543603124735974176 - 0.52582717295145441973270628149009351418*I)^n
+(0.13741867082740688265826177161980396841 + 0.27477457933438246604681642538828208230*I)*(-0.43338019958693104649543603124735974176 + 0.52582717295145441973270628149009351418*I)^n
+(-0.63741867082740688265826177161980396834 - 0.17749243753226296792340167506681155647*I)*(0.93338019958693104649543603124735974176 - 1.1324852222262455048247600868452498278*I)^n
+(-0.63741867082740688265826177161980396834 + 0.17749243753226296792340167506681155647*I)*(0.93338019958693104649543603124735974176 + 1.1324852222262455048247600868452498278*I)^n
除了第一项不满足

点评

wayne
从通项公式的角度来看，第一项不满足，但是从生成函数的角度来看，倒是可以include进来的  发表于 2019-5-11 10:32
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wayne
发表于 2019-5-11 09:50:39 | 显示全部楼层
哈哈哈，我们在玩函数拟合，mathe 直接给精确公式，逆天了！ 反观一下，可见 Fans出题 居心叵测，
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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wayne
发表于 2019-5-11 10:05:32 | 显示全部楼层
额，其实本可以从一开始直接用Mathematica的一个函数FindGeneratingFunction就能搞定，只这么一试，就把生成函数给一股脑搞出来了。
  1. FindGeneratingFunction[data[[All, 2]], x]
复制代码


\[G[x]= \frac{x^2 \left(3 x^4+2 x^3+x^2-4 x+1\right)}{x^5-2 x^2+2 x-1} =-x^2+2 x^3+5 x^4+4 x^5-5 x^6-19 x^7-26 x^8-9 x^9+38 x^{10}+89 x^{11}+83 x^{12}-38 x^{13}-251 x^{14}-388 x^{15}-185 x^{16}+489 x^{17}+1310 x^{18}+1391 x^{19}-226 x^{20}+O\left(x^{21}\right)\]
从生成函数的角度来看，第一个值{1,0}反倒是合理的，莫非是我错怪Fans了
。

然后从生成函数求公式：
  1. ans = SeriesCoefficient[(  x^2 (1 - 4 x + x^2 + 2 x^3 + 3 x^4))/(-1 + 2 x - 2 x^2 + x^5), {x,  0, n}]
复制代码

可以给出通项公式的解析表达，但是排版比较混乱，我就直接给出数值的样子，给出$0 < n < 1 || n > 1$的分支：
\[-1+(0.18291428413576543764+0.41209376567055208885 I) (-0.43338019958693104650-0.52582717295145441973 I)^{n+1}+(0.18291428413576543764-0.41209376567055208885 I) (-0.43338019958693104650+0.52582717295145441973 I)^{n+1}-(0.18291428413576543764+0.41209376567055208885 I) (0.9333801995869310465-1.1324852222262455048 I)^{n+1}-(0.18291428413576543764-0.41209376567055208885 I) (0.9333801995869310465+1.1324852222262455048 I)^{n+1}\]
跟前面mathe的计算应该是一样的。

点评

wayne
是的，这个妖异是人为产生的。  发表于 2019-5-11 11:23
mathe
你的GenerationFunction分子的次数高于分母，就说明前面有一项是妖异的。正常分子的次数应该低于分母。  发表于 2019-5-11 11:16
毋因群疑而阻独见　　毋任己意而废人言
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wayne
发表于 2019-5-11 11:42:49 | 显示全部楼层
临门一脚，所以，本题是$ a*b^n-1$的实部：
其中， a 是 $80 + 88 x^2 + 242 x^3 + 121 x^4 = 0$的根。
\[a = -1.27483734165481376531652354323960793667237183096752 +  0.35498487506452593584680335013362311293032992693839 I\]
b 是方程 $1 + x + x^2 - x^3 + x^4 = 0$的根
\[ b = 0.9333801995869310464954360312473597417565923343049 +  1.1324852222262455048247600868452498277862122474372 I\]
与7楼的函数拟合的结果完美匹配
\[ a = -1.2748373416548137601939656027607260609539677534973 +
0.354984875064525930534409910078275152662157966757 I\]
\[ b =0.9333801995869310465628603640680169309656399643975 +
1.1324852222262455048500390346150772548588801289209 I\]

点评

wayne
^_^  发表于 2019-5-11 13:03
KeyTo9_Fans
好吧，既然是这样的结果，我们只好认为楼主的数列有$2$个bug：第$1$、噪声$-1$是个bug，第$2$、数列的第$1$项也是个bug。  发表于 2019-5-11 12:06
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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wayne
发表于 2019-5-11 13:00:14 | 显示全部楼层
把精确值回代进去，计算一下误差，妥妥的， ：
  1. Table[{n,data[[n]],N[Re[Root[80+88 #^2+242 #^3+121 #^4&,2,0]Root[1+#+#^2-#^3+#^4&,4,0]^n]-1-data[[n,2]],10]},{n,Length[data]}]//Column
复制代码

  1. {{1,0},-2.591923058}
  2. {{2,-1},-0.2260944892}
  3. {{3,2},0.006494478198}
  4. {{4,5},0.09934936662}
  5. {{5,4},-0.08912756481}
  6. {{6,-5},0.03112307962}
  7. {{7,-19},0.01440679961}
  8. {{8,-26},-0.02693808181}
  9. {{9,-9},0.01665960377}
  10. {{10,38},-0.001932193651}
  11. {{11,89},-0.006060515220}
  12. {{12,83},0.006150156475}
  13. {{13,-38},-0.002516738422}
  14. {{14,-251},-0.0006741860254}
  15. {{15,-388},0.001752911142}
  16. {{16,-185},-0.001206320886}
  17. {{17,489},0.0002316924195}
  18. {{18,1310},0.0003592881888}
  19. {{19,1391},-0.0004189944867}
  20. {{20,-226},0.0001963457907}
  21. {{21,-3419},0.00002435966911}
  22. {{22,-5897},-0.0001122798237}
  23. {{23,-3646},0.00008600920319}
  24. {{24,5893},-0.00002241643294}
  25. {{25,18852},-0.00002050548153}
  26. {{26,22499},0.00002818157192}
  27. {{27,1397},-0.00001490571681}
  28. {{28,-45850},-1.653742512*10^-7}
  29. {{29,-88601},7.064252174*10^-6}
  30. {{30,-66650},-6.046228684*10^-6}
  31. {{31,66401},1.960610200*10^-6}
  32. {{32,267499},1.107960961*10^-6}
  33. {{33,356346},-1.870672730*10^-6}
  34. {{34,89093},1.106984793*10^-6}
  35. {{35,-601156},-9.091363751*10^-8}
  36. {{36,-1314097},-4.351866614*10^-7}
  37. {{37,-1158383},4.194149129*10^-7}
  38. {{38,667774},-1.614695814*10^-7}
  39. {{39,3741407},-5.478419543*10^-8}
  40. {{40,5546110},1.224571344*10^-7}
  41. {{41,2295309},-8.070400174*10^-8}
  42. {{42,-7659985},1.309264069*10^-8}
  43. {{43,-19242814},2.612370347*10^-8}
  44. {{44,-19424251},-2.872206986*10^-8}
  45. {{45,5183236},1.276558771*10^-8}
  46. {{46,51510283},2.271313413*10^-9}
  47. {{47,84994109},-7.895907910*10^-9}
  48. {{48,47724838},5.789260826*10^-9}
  49. {{49,-93962793},-1.351732389*10^-9}
  50. {{50,-278192026},-1.516398718*10^-9}
  51. {{51,-316948183},1.941980756*10^-9}
  52. {{52,7481795},-9.791489637*10^-10}
  53. {{53,696584794},-5.299861281*10^-11}
  54. {{54,1284243205},5.005683129*10^-10}
  55. {{55,897124796},-4.092648663*10^-10}
  56. {{56,-1091185001},1.223143974*10^-10}
  57. {{57,-3969137799},8.400956362*10^-11}
  58. {{58,-5059320802},-1.296082804*10^-10}
  59. {{59,-896122801},7.333262483*10^-11}
  60. {{60,9223520798},-3.383055808*10^-12}
  61. {{61,19148102197},-3.111696387*10^-11}
  62. {{62,15880024999},2.854174751*10^-11}
  63. {{63,-11595475198},-1.029085765*10^-11}
  64. {{64,-55847123195},-4.332585483*10^-12}
  65. {{65,-79279775196},8.533488531*10^-12}
  66. {{66,-27717201805},-5.384815838*10^-12}
  67. {{67,119005171781},7.051387657*10^-13}
  68. {{68,281849271974},1.889051555*10^-12}
  69. {{69,269841077191},-1.964759903*10^-12}
  70. {{70,-103296164762},8.258656138*10^-13}
  71. {{71,-773991685711},1.964351959*10^-13}
  72. {{72,-1222385870117},-5.537220701*10^-13}
  73. {{73,-614939096838},3.887370234*10^-13}
  74. {{74,1484734623749},-7.984171620*10^-14}
  75. {{75,4096051276412},-1.112918654*10^-13}
  76. {{76,4448641619615},1.335348976*10^-13}
  77. {{77,-517205183711},-6.406854426*10^-14}
  78. {{78,-10546632703490},-6.469860237*10^-15}
  79. {{79,-18574120415809},3.535565185*10^-14}
  80. {{80,-11958924148226},-2.764084120*10^-14}
  81. {{81,17679034154781},7.541911448*10^-15}
  82. {{82,58758711422303},6.296961031*10^-15}
  83. {{83,71612721831554},-8.959761070*10^-15}
  84. {{84,7133900402693},4.842207650*10^-15}
  85. {{85,-140916567005948},-3.690375935*10^-17}
  86. {{86,-278421900662501},-2.216311370*10^-15}
  87. {{87,-216251955890803},1.938145809*10^-15}
  88. {{88,195952611374950},-6.508467111*10^-16}
  89. {{89,831543034934199},-3.357773903*10^-16}
  90. {{90,1130264280112550},5.932348821*10^-16}
  91. {{91,319020589694201},-3.582868256*10^-16}
  92. {{92,-1838739336727501},3.510239366*10^-17}
  93. {{93,-4119567241468454},1.359317275*10^-16}
  94. {{94,-3730112774547707},-1.341187226*10^-16}
  95. {{95,1909173213954044},5.313398187*10^-17}
  96. {{96,11597592566697703},1.621858322*10^-17}
  97. {{97,17538099368759817},-3.872840364*10^-17}
  98. {{98,7761446362655774},2.603775384*10^-17}
  99. {{99,-23283418786755793},-4.586407613*10^-18}
  100. {{100,-60180557084869090},-8.114341034*10^-18}
  101. {{101,-62196684029528891},9.162716377*10^-18}
  102. {{102,13505845479440215},-4.174288815*10^-18}
  103. {{103,159166505380593986},-6.362565458*10^-19}
  104. {{104,268037901015551749},2.489656925*10^-18}
  105. {{105,157562234185046436},-1.862514091*10^-18}
  106. {{106,-283148017690539517},4.583743440*10^-19}
  107. {{107,-867914658271731691},4.674880554*10^-19}
  108. {{108,-1010366775781790362},-6.180291230*10^-19}
  109. {{109,-16866334004565593},3.186225687*10^-19}
  110. {{110,2144563117739495974},1.078929230*10^-20}
  111. {{111,4039710885797583617},-1.572922088*10^-19}
  112. {{112,2922380877844443595},1.313250532*10^-19}
  113. {{113,-3245026791688070406},-4.079459901*10^-20}
  114. {{114,-12351681673069593595},-2.561673570*10^-20}
  115. {{115,-16068746645023550404},4.114501893*10^-20}
  116. {{116,-3394419058110330001},-2.376869954*10^-20}
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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