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单词 Quadratic Recurrence
释义

Quadratic Recurrence

N.B. A detailed on-line essay by S. Finchwas the starting point for this entry.


A quadratic recurrence is a Recurrence Relation on a Sequence of numbers expressing as asecond degree polynomial in with . For example,

(1)

is a quadratic recurrence. Another simple example is
(2)

with , which has solution . Another example is the number of ``strongly'' binary trees of height, given by
(3)

with . This has solution
(4)

where
(5)

and is the Floor Function (Aho and Sloane 1973). A third example is the closest strict underapproximation ofthe number 1,
(6)

where are integers. The solution is given by the recurrence
(7)

with . This has a closed solution as
(8)

where


(9)

(Aho and Sloane 1973). A final example is the well-known recurrence
(10)

with used to generate the Mandelbrot Set.

See also Mandelbrot Set, Recurrence Relation


References

Aho, A. V. and Sloane, N. J. A. ``Some Doubly Exponential Sequences.'' Fib. Quart. 11, 429-437, 1973.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/quad/quad.html
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更新时间：2025/4/6 10:57:01