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Summing a family of generalized Pell numbers. (English) Zbl 1471.11077

Summary: A new family of generalized Pell numbers was recently introduced and studied by D. Bród [Ann. Math. Sil. 33, 66–76 (2019; Zbl 1470.11022)]. These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power \(P^\ell_n\) is expressed as a linear combination of \(P_{mn}\). The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter \(R=2^r\), the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
05A15 Exact enumeration problems, generating functions

Citations:

Zbl 1470.11022

Software:

OEIS
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Full Text: DOI arXiv

References:

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