Fibonacci numbers and the Pascal Triangle

 

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Cassini formula for Fibonacci numbers

        

Giovanni Cassini. Born: 8 June 1625 in Perinaldo, Republic of Genoa (now Italy). Died: 14 Sept 1712 in Paris, France.We know little of his parents but certainly his father was a Tuscan. In fact Giovanni was brought up, not by his parents but by an uncle, a brother of his mother Julia Crovesi. After spending two years being educated at Vallebone, Cassini entered the Jesuit College at Genoa where he studied under Casselli. After this he studied at the abbey of San Fructuoso. He showed great intellectual curiosity and was especially interested in poetry, mathematics and astronomy. His first interest, however, was in astrology rather than astronomy. He read widely on this topic and soon was very knowledgeable, yet was convinced that there was no truth in astrological predictions. It was, rather strangely, his extensive knowledge of astrology that led to his first appointment. In 1644 the Marquis Cornelio Malvasia, who was a senator from Bologna with a great interest in astrology, invited Cassini to Bologna. He offered him a position in the Panzano Observatory which he was constructing at that time. Cassini was an astronomer at the Panzano Observatory, from 1648 to 1669. He was a professor of astronomy at the University of Bologna. Cassini's brilliant discoveries gave him an international reputation and led to him being invited to Paris by Louis XIV in 1668.

He thoroughly adopted his new country, to the extent that he became interchangeably known as Jean-Dominique Cassini. Cassini became,in 1671, director of the Paris Observatory.Cassini would remain the director of it for the rest of his career until his death in 1712. Along with Hooke, Cassini is given credit for the discovery of the Great Red Spot (~1665). Cassini was the first to observe four of Saturn's moons; he also discovered the Cassini Division (1675). Around 1690, Cassini was the first to observe differential rotation within Jupiter's atmosphere. In 1672 he sent his colleague Jean Richer to Cayenne, French Guiana, while he himself stayed in Paris. The two made simultaneous observations of Mars and thus found its parallax to determine its distance, thus measuring for the first time the true dimensions of the solar system. Cassini was the first to make successful measurements of longitude by the method suggested by Galileo, using eclipses of the satellites of Jupiter as a clock. In 1680 he studied the Cassinian curve which is the locus of a point the product of whose distances from two fixed foci is constant. He worked on this as part of a study of the relative motions of the Earth and the sun and proposed this as the curve for planetary orbits rather than the ellipse as proposed by Kepler.

The original Cassini formula for Fibonacci numbers is :
Here, we shall give a proof for the Giovanni Cassini`s formula:

Well-known Binet`s formula for Lucas numbers that can be used to prove Cassini formula is:



from which we get Binet`s operator equation for Fibonacci and Lucas numbers:



We shell use the next Fibonacci formula ( Vajda-11,Dunlap-7, Lucas-1876):



However, following the operator`s calculus, we obtain the Cassini formula for Fibonacci numbers:



Many of The Fibonacci and Lucas sequences proprties follow immediately from the recursive rule that each term is the sum of the two preceding terms. Similarly, it is easy to establish corresponding form for Cassini Lucas numbers formula.
Here, we shall give a proof for the Cassini`s Lucas formula by using general Fibonacci and Lucas Numbers relation ( Vajda-20b ):



Using the context of Difference operators and the Fibonacci space, we can derive useful results.

     

        

  2001-2005 Radoslav Jovanovic                 updated:  January 2005.