소모스 수열(Somos sequence)

이 항목의 스프링노트 원문주소

• 소모스 수열(Somos sequence)

 

 

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) [Robinson1992]
  

 

 

 

소모스-4 수열

• \(a_{n+4}a_{n} = a_{n+3} a_{n+2} + a_{n+1}^2\)
  
• 초기조건 \(a_1=a_2=a_3=a_4=1\) 인 경우
  
• 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}]
   

• http://oeis.org/A006720
  
• http://oeis.org/A006769
  
• 초기조건이 \(a_1=x,a_2=y,a_3=z,a_4=w\) 인 경우
  \(x,y,z,w,\frac{w y+z^2}{x},\frac{w^2 x+w y z+z^3}{x y},\frac{y(wy+z^2)^2+w x (w^2 x+w y z+z^3)}{x^2 y z}\)
  
• 이를 로랑현상(Laurent phenomenon) 이라 한다
  

1. RecurrenceTable[{a[n] a[n - 4] == a[n - 1] a[n - 3] + a[n - 2]^2,
     a[1] == x, a[2] == y, a[3] == z, a[4] == w}, a, {n, 10}]
   

 

 

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}]
   

• http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.1.html
• http://oeis.org/A006721
  

 

 

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[1]
  

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}]
   

• http://cis.csuohio.edu/~somos/somos6.html
  

• http://oeis.org/A006722
  

 

 

소모스-8 수열

• \(a_{n+8}a_{n} = a_{n+7} a_{n+1} +a_{n+6}a_{n+2}+a_{n+5}a_{n+3}+a_{n+4}^2\)
  
• 1,1,1,1,1,1,1,1,4,7,13,25,61,187,775,5827,14815,420514/7,28670773/91,6905822101/2275
  

1. RecurrenceTable[{a[n] a[n - 8] ==    a[n - 1] a[n - 7] + a[n - 2] a[n - 6] + a[n - 3] a[n - 5] +
       a[n - 4]^2, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == 1,
      a[6] == 1, a[7] == 1, a[8] == 1}, a, {n, 20}]
   

 

재미있는 사실

 

• Math Overflow http://mathoverflow.net/search?q=
• 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=

 

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• Earliest Known Uses of Some of the Words of Mathematics
• Earliest Uses of Various Mathematical Symbols
• 수학사연표

 

 

메모

• http://www.cut-the-knot.org/arithmetic/algebra/SimpleSomosSequence.shtml
• http://faculty.uml.edu/jpropp/somos.html
• http://www.math.brown.edu/~jhs/Presentations/ICMSEDSLecture.pdf

 

 

 

수학용어번역

• 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
• 발음사전 http://www.forvo.com/search/
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

 

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• [2]http://en.wikipedia.org/wiki/Somos_sequence
• http://en.wikipedia.org/wiki/Elliptic_divisibility_sequence
• http://www.proofwiki.org/wiki/
• http://www.wolframalpha.com/input/?i=
• The Online Encyclopaedia of Mathematics
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences

 

 

관련논문

• Hone, Andrew N. W. 2010. Analytic solutions and integrability for bilinear recurrences of order six. Applicable Analysis: An International Journal 89, no. 4: 473. doi:10.1080/00036810903329977. 
• Hone, A. N. W. 2007. Sigma function solution of the initial value problem for Somos 5 sequences doi:0.1090/S0002-9947-07-04215-8
• 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:10.1112/S0024609304004163. 
• 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
• van der Poorten, Alfred J. 2004. Elliptic curves and continued fractions. math/0403225 (March 14). http://arxiv.org/abs/math/0403225.
• [Robinson1992]R. M. Robinson, "Periodicity of Somos sequences", Proc. Amer. Math. Soc., 116 (1992), 613-619. doi:10.1090/S0002-9939-1992-1140672-5
• 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.ams.org/mathscinet
• http://dx.doi.org/10.1090/S0002-9947-07-04215-8

 

 

관련도서

• Gale, David. 1998. Tracking The Automatic Ant: And Other Mathematical Explorations. Springer, May 29. 
   
  
• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/mainSearch.do?query=
  • http://book.daum.net/search/mainSearch.do?query=

 

 

관련기사

• 네이버 뉴스 검색 (키워드 수정)
  
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=

 

 

링크

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
• 네이버 오늘의과학
• Mathematical Moments from the AMS
• BetterExplained
• exampleproblems.com