We will see in this chapter how Fibonacci numbers can be extracted from Pascal's triangle. In addition, we will derive a host of Fibonacci and Lucas identities involving binomial coefficients.
We will begin with a brief discussion of binomial coefficients, which are coefficients occurring in the binomial expansion of . The term binomial coefficient was introduced by the greatest German algebraist Michel Stifel (1486–1567).
The earliest known occurrence of binomial coefficients can be traced back to a tenth-century commentary by the Indian mathematician Halayudha on Pingala's Chandas Shastra; interestingly, it also contains a discourse on what is now called Pascal's triangle (called Meru Prastara). Bhaskara (1114–1185?) gives a concise discussion of binomial coefficients in his 1150 classic work Leelavati.
11.1 BINOMIAL COEFFICIENTS
Let n and k be nonnegative integers. The binomial coefficient is defined by
It is also denoted by and . Read as “n choose ...