"소모스 수열(Somos sequence)"의 두 판 사이의 차이

수학노트
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18번째 줄: 18번째 줄:
 
 
 
 
  
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<h5 style="margin: 0px; line-height: 2em;">소모스 4-수열</h5>
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<h5 style="margin: 0px; line-height: 2em;">소모스-4 수열</h5>
  
 
* <math>a_{n+4}a_{n} = a_{n+3} a_{n+2} + a_{n+1}^2</math><br>
 
* <math>a_{n+4}a_{n} = a_{n+3} a_{n+2} + a_{n+1}^2</math><br>
33번째 줄: 33번째 줄:
 
 
 
 
  
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<h5 style="margin: 0px; line-height: 2em;">소모스 5-수열</h5>
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<h5 style="margin: 0px; line-height: 2em;">소모스5- 수열</h5>
  
 
* <math>a_{n+5}a_{n} = a_{n+4} a_{n+1} + a_{n+3} a_{n+2}</math><br>
 
* <math>a_{n+5}a_{n} = a_{n+4} a_{n+1} + a_{n+3} a_{n+2}</math><br>
41번째 줄: 41번째 줄:
  
 
* [http://www-groups.dcs.st-and.ac.uk/%7Ejohn/Zagier/Solution5.1.html ][http://www-groups.dcs.st-and.ac.uk/%7Ejohn/Zagier/Solution5.1.html http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.1.html]
 
* [http://www-groups.dcs.st-and.ac.uk/%7Ejohn/Zagier/Solution5.1.html ][http://www-groups.dcs.st-and.ac.uk/%7Ejohn/Zagier/Solution5.1.html http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.1.html]
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* http://oeis.org/A006721<br>
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<h5 style="margin: 0px; line-height: 2em;">소모스-6 수열</h5>
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* <math>a_{n+6}a_{n} = a_{n+5} a_{n+1} +a_{n+4}a_{n+2}+ a_{n+3}^2</math><br>
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*  1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875[http://oeis.org/A006722 ]<br>
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#  RecurrenceTable[{a[n] a[n - 6] == a[n - 1] a[n - 5] + a[n - 2] a[n - 4] + a[n - 3]^2, a[1] == 1,   a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 20}]<br>
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* http://oeis.org/A006722<br>
  
 
 
 
 
115번째 줄: 129번째 줄:
  
 
* Hone, A. N. W. 2005. Elliptic Curves and Quadratic Recurrence Sequences. Bulletin of the London Mathematical Society 37, no. 2 (April 1): 161 -171. doi:[http://dx.doi.org/10.1112/S0024609304004163 10.1112/S0024609304004163]. 
 
* Hone, A. N. W. 2005. Elliptic Curves and Quadratic Recurrence Sequences. Bulletin of the London Mathematical Society 37, no. 2 (April 1): 161 -171. doi:[http://dx.doi.org/10.1112/S0024609304004163 10.1112/S0024609304004163]. 
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* Swart, Christine, and Andrew Hone. 2005. Integrality and the Laurent phenomenon for Somos 4 sequences. math/0508094 (August 4). http://arxiv.org/abs/math/0508094<br>  <br>
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* Swart, Christine, and Andrew Hone. 2005. Integrality and the Laurent phenomenon for Somos 4 sequences. math/0508094 (August 4). http://arxiv.org/abs/math/0508094
 
* R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.
 
* R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619.
 
* David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequences", Math. Intelligencer, 13(1) (1991), pp. 40-42.
 
* David Gale, Mathematical Entertainments: "The strange and surprising saga of the Somos sequences", Math. Intelligencer, 13(1) (1991), pp. 40-42.
 
* http://www.jstor.org/action/doBasicSearch?Query=
 
* http://www.jstor.org/action/doBasicSearch?Query=
 
* http://www.ams.org/mathscinet
 
* http://www.ams.org/mathscinet
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* http://dx.doi.org/
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* http://dx.doi.org/10.1090/S0002-9939-1992-1140672-5
  
 
 
 
 

2011년 2월 25일 (금) 05:21 판

이 항목의 스프링노트 원문주소

 

 

개요
  • 소모스 4,5,6,7 은 정수수열
  • 소모스 8,9는 정수수열이 아니다
  • 정수수열이 되는가의 문제 (integrality)
  • 합동식을 생각할 때의 주기성 문제 (periodicity modulo n)

 

 

 

소모스-4 수열
  • \(a_{n+4}a_{n} = a_{n+3} a_{n+2} + a_{n+1}^2\)
  • 1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786
  1. RecurrenceTable[{a[n] a[n - 4] == a[n - 1] a[n - 3] + a[n - 2]^2,  a[1] == 1, a[2] == 1,   a[3] == 1, a[4] == 1}, a,    {n, 20}]

 

 

 

소모스5- 수열
  • \(a_{n+5}a_{n} = a_{n+4} a_{n+1} + a_{n+3} a_{n+2}\)
  • 1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933
  1. RecurrenceTable[{a[n] a[5 + n] == a[2 + n] a[3 + n] + a[1 + n] a[4 + n], a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1}, a,   {n, 20}]

 

 

소모스-6 수열
  • \(a_{n+6}a_{n} = a_{n+5} a_{n+1} +a_{n+4}a_{n+2}+ a_{n+3}^2\)
  • 1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875[2]
  1. RecurrenceTable[{a[n] a[n - 6] == a[n - 1] a[n - 5] + a[n - 2] a[n - 4] + a[n - 3]^2, a[1] == 1,   a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1, a[6] == 1}, a, {n, 20}]

 

 

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