“Somos Sequence”的概念、定义、习题解题方法及翻译-数学辞典

单词

Somos Sequence

释义

Somos Sequence

The Somos sequences are a set of related symmetrical Recurrence Relations which,surprisingly, always give integers. The Somos sequence of order is defined by

where

is the Floor Function and

for

, ...,

. The 2- and 3-Somossequences consist entirely of 1s. The

-Somos sequences for

, 5, 6, and 7 are

giving 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, ... (Sloane's A006720),1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, ... (Sloane's A006721),1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, ... (Sloane's A006722),1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, ... (Sloane's A006723).Gale (1991) gives simple proofs of the integer-only property of the 4-Somos and 5-Somos sequences. Hickerson proved6-Somos generates only integers using computer algebra, and empirical evidence suggests 7-Somos is also integer-only.

However, the -Somos sequences for do not give integers. The values of for which firstbecomes nonintegral for the -Somos sequence for , 9, ... are17, 19, 20, 22, 24, 27, 28, 30, 33, 34, 36, 39, 41, 42, 44, 46, 48, 51, 52, 55, 56, 58, 60, ...(Sloane's A030127).

See also Göbel's Sequence, Heronian Triangle

References

Buchholz, R. H. and Rathbun, R. L. ``An Infinite Set of Heron Triangles with Two Rational Medians.'' Amer. Math. Monthly 104, 107-115, 1997.

Gale, D. ``Mathematical Entertainments: The Strange and Surprising Saga of the Somos Sequences.'' Math. Intel. 13, 40-42, 1991.

Sloane, N. J. A. SequencesA030127,A006720/M0857,A006721/M0735,A006722/M2457,A006723/M2456in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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