Geometric structure of generalized Fibonacci vectors

Đặng Võ Phúc

doi:10.56754/0719-0646.2803.577

Authors

Đặng Võ Phúc Department of Mathematics, FPT University, An Phu Thinh New Urban Area, Quy Nhon, Vietnam https://orcid.org/0000-0002-6885-3996

MSC:

11B39, 15A63, 11B83

DOI:

https://doi.org/10.56754/0719-0646.2803.577

Keywords:

linear recurrence, Lucas sequences, consecutive-term vectors, inner product, exterior product, Pell numbers

Abstract

Consecutive-term vectors associated with second-order linear recurrences have been studied in several settings, including generalized Fibonacci vectors, Fibonacci/Lucas vectors, Horadam vectors, and Pell–Pell/Lucas vectors. In this paper, we give a unified geometric formulation for length-\(m\) vectors generated by the recurrence \(U_{n}=pU_{n-1}+qU_{n-2}\) under the hypothesis of two distinct real characteristic roots. Using the intrinsic two-dimensional structure of the solution space, we derive Binet-type vector formulas and show that every exterior product of two consecutive-term vectors is a scalar multiple of one fixed bivector. This exterior-algebra formulation explains the planar confinement observed in three dimensions and yields closed-form inner products in arbitrary dimension, including mixed products involving the normalized Lucas-type companion sequence. The Pell specialization recovers the consistent Pell/Pell–Lucas vector identities and makes their rank-two origin explicit.

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