FENCE TILING DERIVED IDENTITIES INVOLVING THE METALLONACCI NUMBERS SQUARED OR CUBED
5
Issued Date
2023-12-01
Resource Type
ISSN
00150517
Scopus ID
2-s2.0-85164435567
Journal Title
Fibonacci Quarterly
Volume
60
Issue
5
Start Page
5
End Page
17
Rights Holder(s)
SCOPUS
Bibliographic Citation
Fibonacci Quarterly Vol.60 No.5 (2023) , 5-17
Suggested Citation
Allen M.A. FENCE TILING DERIVED IDENTITIES INVOLVING THE METALLONACCI NUMBERS SQUARED OR CUBED. Fibonacci Quarterly Vol.60 No.5 (2023) , 5-17. 17. Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/87990
Title
FENCE TILING DERIVED IDENTITIES INVOLVING THE METALLONACCI NUMBERS SQUARED OR CUBED
Author(s)
Author's Affiliation
Other Contributor(s)
Abstract
We refer to the generalized Fibonacci sequence (Mn(c))n≥0, where Mn(c+1) = cMn(c) + Mn(c−)1 for n > 0 with M0(c) = 0, M1(c) = 1, for c = 1, 2, . . . as the c-metallonacci numbers. We consider the tiling of an n-board (an n × 1 rectangular board) with c colours of 1/p × 1 tiles (with the shorter sides always aligned horizontally) and (1/p, 1 − 1/p)-fence tiles for p ∈ Z+. A (w, g)-fence tile is composed of two w × 1 sub-tiles separated by a g × 1 gap. The number of such tilings equals (Mn(c+1) )p and we use this result for the cases p = 2, 3 to devise straightforward combinatorial proofs of identities relating the metallonacci numbers squared or cubed to other combinations of metallonacci numbers. Special cases include relations between the Pell numbers cubed and the even Fibonacci numbers. Most of the identities derived here appear to be new.