Abstract
The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.
Highlights
The defined of second-order linear recurrence sequence {Sn } is
We extend the recursive property of Sn to all negative integers
When x = 1, Fn (1)√= Fn becomes known as the Fibonacci sequence
Summary
The defined of second-order linear recurrence sequence {Sn } is Yi Yuan and Zhang Wenpeng [2] proved the following conclusion: For any positive integer n and k, one has the identity a1 + a2 +···+ ak =n Ma Yuankui and Zhang Wenpeng [3] studied this problem, and proved the following result: 1 h (−1)h− j · S(h, j) x2h− j a1 + a2 +···+ ah+1 =n
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