소모스 수열(Somos sequence)

수학노트
Pythagoras0 (토론 | 기여) 사용자의 2015년 12월 25일 (금) 01:34 판 (관련논문)

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개요

  • 점화식으로 정의되는 수열
  • 소모스 4,5,6,7 은 정수수열이며 소모스 8,9는 정수수열이 아니다
  • 점화식만으로는 정수수열이 되는가가 자명하지 않다
  • 정수수열이 되는가의 문제 (integrality)
  • 합동식을 생각할 때의 주기성 문제 (periodicity modulo n) [Robinson1992]
  • 타원곡선론과 클러스터 대수(cluster algebra) 등의 이론에서 등장



소모스-4 수열

  • \(a_{n+4}a_{n} = a_{n+3} a_{n+1} + a_{n+1}^2\)



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[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}]



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}]



역사



메모


관련된 항목들

사전 형태의 자료



관련논문

  • Fedorov, Yuri N., and Andrew N. W. Hone. “Sigma-Function Solution to the General Somos-6 Recurrence via Hyperelliptic Prym Varieties.” arXiv:1512.00056 [nlin], November 30, 2015. http://arxiv.org/abs/1512.00056.
  • Gorman, Alexi Block, Tyler Genao, Heesu Hwang, Noam Kantor, Sarah Parsons, and Jeremy Rouse. “The Density of Primes Dividing a Particular Non-Linear Recurrence Sequence.” arXiv:1508.02464 [math], August 10, 2015. http://arxiv.org/abs/1508.02464.
  • Davis, Bryant, Rebecca Kotsonis, and Jeremy Rouse. “The Density of Primes Dividing a Term in the Somos-5 Sequence.” arXiv:1507.05896 [math], July 21, 2015. http://arxiv.org/abs/1507.05896.
  • 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
  • R.W. Gosper, R. Schroeppel, Somos sequence near-addition formulas and modular theta functions, arXiv:math.NT/0703470v1, 15 March 2007.
  • 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). [2]http://arxiv.org/abs/math/0403225.
  • [FZ2001]Fomin, Sergey, and Andrei Zelevinsky. 2001. The Laurent phenomenon. math/0104241 (April 25). http://arxiv.org/abs/math/0104241.
  • [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.

관련도서

  • Gale, David. 1998. Tracking The Automatic Ant: And Other Mathematical Explorations. Springer, May 29.