Wamiliana, Wamiliana and Suharsono, Suharsono and Kristanto, P.E (2019) Counting the sum of cubes for Lucas and Gibonacci Numbers. Science and Technology Indonesia, 4 (2). pp. 31-35. ISSN e -ISSN:2580-4391 p -ISSN:2580-4405

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Abstract

Lucas and Gibonacci numbers are two sequences of numbers derived from a welknown numbers, Fibonacci numbers. The di�erence between Lucas and Fibonacci numbers only lies on the first and second elements. The first element in Lucas numbers is 2 and the second is 1, and nth element, n ≥ 3 determined by similar pa�ern as in the Fibonacci numbers, i.e : Ln = Ln-1 + Ln-2. Gibonacci numbers G_0,G_1,G_2,G_3. . . ; G_n=G_(n-1)+G_(n-2) are generalized of Fibonacci numbers, and those numbers are nonnegative integers. If G_0=1 and G_1=1, then the numbers are the wellknown Fibonacci numbers, and if G_0=2 and G_1=1, the numbers are Lucas numbers. Thus, the di�erence of those three sequences of numbers only lies on the first and second of the elements in the sequences. For Fibonacci numbers there are quite a lot identities already explored, including the sum of cubes, but there have no discussions yet about the sum of cubes for Lucas and Gibonacci numbers. In this study the sum of cubes of Lucas and Gibonacci numbers will be discussed and showed that the sum of cubes for Lucas numbers is ∑ n i=0 (Li) 3 = L n (L n+1 ) 2 +5(−1) n L n−1 + 19 2 and for Gibonacci numbers is ∑ n i=0 (Gi) 3 = [G n (G n+1 ) 2 +(G 1 −G 0 ) 3 −3G 0 2 G 1 +4G 0 3 −(−1) n (G 1 2 −G 1 G 0 −G 0 2 )G n−1 ] 2 Keywords Fibonacci numbers, Lucas numbers, Gibonacci numbers, identities, sum of cubes

Item Type: Article
Subjects: Q Science > QA Mathematics
Divisions: Fakultas Matematika dan Ilmu Pengetahuan Alam (FMIPA) > Prodi Matematika
Depositing User: Dr Suharsono S
Date Deposited: 08 May 2019 07:33
Last Modified: 08 May 2019 07:33
URI: http://repository.lppm.unila.ac.id/id/eprint/11932

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