Fibonacci Sequence has formed the groundwork of Mathematicians for research in Number Theory. Fibonacci has a profound influence all around us. There are many interesting identities involving Fibonacci. With the invention of Binet’s Formula which is the most fundamental formula for finding nth Fibonacci , further search for generalisation of Fibonacci sequence has almost came to an end but it is interesting to know that Fibonacci numbers can be expressed by Binomial Coefficients. Thus, I tried to convey the process of generalisation of Fibonacci using Binomial coefficients. Arka Mehatari "Generalization of Fibonacci Numbers with Binomial Coefficients and Figurate Numbers" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-5 , August 2021, URL: https://www.ijtsrd.com/papers/ijtsrd46301.pdf Paper URL: https://www.ijtsrd.com/mathemetics/discrete-mathematics/46301/generalization-of-fibonacci-numbers-with-binomial-coefficients-and-figurate-numbers/arka-mehatari
International Journal of Trend
Volume 5 Issue 5, July-August
@ IJTSRD | Unique Paper ID – IJTSRD
Generalization
Binomial Coefficients
Student, St. Micha
ABSTRACT
Fibonacci
Sequence
has
formed
the
groundwork
of
Mathematicians for research in Number Theory.
profound influence all around us. There are many interesting
identities involving Fibonacci. With the invention of Binet’s
Formula (which is the most fundamental formula for finding nth
Fibonacci), further search for generalisation of Fibon
has almost came to an end but it is interesting to know that
Fibonacci numbers can be expressed by Binomial Coefficients.
Thus, I tried to convey the process of generalisation of Fibonacci
using Binomial coefficients.
KEYWORDS:
Introduction and Application of Fibonacci,
Fibonacci Numbers from Pascal’s Triangle, Fibonacci Numbers
using Binomial Coefficients, Fibonacci Numbers from Figurate
Numbers, Some Interesting Sums Involving Fibonacci
I.
INTRODUCTION
Among the greatest Mathematicians of the Middle
Ages Leonardo of Pisa is considered to be one of
the greatest Mathematician. He is mainly known for
his work on Fibonacci. Today, most of us are
acquainted with the Fibonacci Sequence.
The
Fibonacci
Sequence
1,1,2,3,5,8,13,21…
This sequence has a profound influence around us
ranging from the spirals of sunflower, cones of pine
tree and breeding of rabbits.
Fibonacci Sequence has a wide application in
Probability, Combinatorics, Continued Fractions,
Solving Pell’s Equation etc.
Fibonacci gives answer to the problems like,
1. Finding the probability of colouring a n
house with the precondition that no two
adjacent storey has the same colour.
2. Let’s suppose you have to calculate the number
of ways in which you have to pay multiples of
Rs.25 using only Rs.25 and Rs.50.
in Scientific Research and Development
2021 Available Online: www.ijtsrd.com e
46301 | Volume – 5 | Issue – 5 | Jul-Aug
of Fibonacci Numbers with
and Figurate Numbers
Arka Meh