The infinite sum of the cubes of reciprocal Pell numbers
© Xu and Wang; licensee Springer 2013
Received: 17 May 2013
Accepted: 4 June 2013
Published: 26 June 2013
Given the sequence of Pell numbers , we evaluate the integral part of the reciprocal of the sum explicitly in terms of the Pell numbers themselves.
For example, the first few values of and are , .
where is the floor function, that is, it denotes the greatest integer less than or equal to x.
with the initial conditions , , .
Using the method in  seems to be very difficult to deal with for all integers .
for all integers . At the end of , the authors asked whether there exists a corresponding formula for .
In fact, this problem is difficult because it is quite unclear a priori what the shape of the result might be. In order to resolve the question, we carefully applied the method of undetermined coefficients and constructed a number of delicate inequalities in order to complete a proof. The result is as follows.
It remains a difficult problem even to conjecture what might be an analogous expression to the formula for in the theorem for when .
2 Proof of the theorem
for all integers . So, inequalities (3), (4) and (5) hold for all integers .
Combining (6) and (9), we may immediately deduce inequality (2).
for all integers . So, inequalities (11), (12) and (13) hold for all integers .
Combining (14) and (17), we may immediately deduce inequality (10).
Now our theorem follows from inequalities (2) and (10). This completes the proof of our theorem.
The authors express their gratitude to the referee for his very helpful and detailed comments. This work is supported by the N.S.F. (11001218, 11071194) of P.R. China and the Research Fund for the Doctoral Program of Higher Education (20106101120001) of P.R. China and the G.I.C.F. (YZZ12065) of NWU.
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