On Generalized Fibonacci like Sequence Containing Higher Powers

Krishna Kumar Sharma

Abstract


In mathematical terms, sequence is a list of numbers arranged in a specific order. Fibonacci sequence [3] is sequence of whole numbers defined by the recurrence formula Fn=Fn-1 + Fn+2, n>=2 and F0=0, F1=1 where Fn is an nth number of sequence. Fibonacci numbers and their generalization have many interesting and surprising properties. Many authors have defined Fibonacci pattern-based sequences which are popularized and known as Fibonacci-Like sequences. When the first two terms of the Fibonacci sequence become arbitrary, it is known as Fibonacci-like sequence. Fibonacci-like sequence can start at any desired number. This study aimed to derive and validate a formula in solving Fibonacci-like sequence. Here we introduced a generalized Fibonacci Like sequence defined by Gn+1 + bGn-1, n>=1 and G0=0, G1=s where a, b and s are real numbers. In this research paper we introduced and derived various analytical properties. These identities of Generalized Fibonacci like sequence containing higher powers have been proved by Binet’s formula and other techniques.  

 


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References


Brousseau B A, “Linear Recursion Relations Lesson Three, The Binet Formula” The Fibonacci Quarterly 7 (1969) 99-104.

Carlitz L & Hunter J A H, “Sum of Power of Fibonacci and Lucas Numbers” The Fibonacci Quarterly 7 (1969) 467-473.

Coshy T, “Fibonacci and Lucas Numbers with Applications”, Wiley-Interscience Publication New York. (2001).

Falcon S, and Plaza A, “On the Fibonacci K Numbers”, Chaos Solution and Fractals, Vol 32(5)2007, 1615-1624.

Gupta Y., Sikhwal O and Sisodiya K, “Determinantal Identities of Generalized Fibonacci-Like Sequence”, MAYFEB Journal of Computer Science, Vol. 1 (2017) 1-6.

Gupta Y.K, Singh M, Sikhwal O, “Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences”, Turkish Journal of Analysis and Number Theory, Vol. 2, No. 6 (2014), 233-238.

Natividad L.R, “Deriving a Formula in Solving Fibonacci Like Sequence”, International Journal of Mathematics and Scientific Company, 1(2010), 19-21.

Singh B & Sikhwal O, “Generalized Fibonacci Sequence and Analytical Properties” Vikram Mathematics Journal, 26 (2006), 131-144.

Singh M ,Sikhwal O and Gupta Y K, “ Identities of Generalized Fibonacci like Sequence”, Turkish Journal of Analysis and Number Theory, Vol. 2, No. 5 (2014), 170-175.

Waddil M E and Sacks-L, “Another generalized Fibonacci sequence, The Fibonacci Quarterly 5 (1967) 209-222.

Yalavigi C C, “B -169 Elementary Problem” The Fibonacci Quarterly 7 (1969) 332.


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MAYFEB Journal of Mathematics 
Toronto, Ontario, Canada
MAYFEB TECHNOLOGY DEVELOPMENT
ISSN 2371-6193