Applications of Fibonacci numbers. : Volume 9 proceedings of by Fredric T. Howard

By Fredric T. Howard

A document at the 10th foreign convention. Authors, coauthors and different convention contributors. Foreword. The organizing committees. record of individuals to the convention. advent. Fibonacci, Vern and Dan. common Bernoulli polynomials and P-adic congruences; A. Adelberg. A generalization of Durrmeyer-type polynomials and their approximation houses; O. Agratini. Fibinomial identities; A.T. Benjamin, J.J. Quinn, J.A. Rouse. Recounting binomial Fibonacci identities; A.T. Benjamin, J.A. Rouse. The Fibonacci diatomic array utilized to Fibonacci representations; M. Bicknell-Johnson. discovering Fibonacci in a fractal; N.C. Blecke, okay. Fleming, G.W. Grossman. confident integers (a2 + b2) / (ab + 1) are squares; J.-P. Bode, H. Harborth. at the Fibonacci size of powers of dihedral teams; C.M. Campbell, P.P. Campbell, H. Doostie, E.F. Robertson. a few sums on the topic of sums of Oresme numbers; C.K. prepare dinner. a few ideas on rook polynomials on sq. chessboards; D. Fielder. Pythagorean quadrilaterals; R. Hochberg, G. Hurlbert. A normal lacunary recurrence formulation; F.T. Howard. Ordering phrases and units of numbers: the Fibonacci case; C. Kimberling. a few uncomplicated houses of a Tribonacci line-sequence; J.Y. Lee. one of those series comprised of Fibonacci numbers; Aihua. Li, S. Unnithan. Cullen numbers in binary recurrent sequences; F. Luca, P. Stanica. A generalization of Euler's formulation and its connection to Bonacci numbers; J.F. Mason, R.H. Hudson. Extensions of generalized binomial coefficients; R.L. Ollerton, A.G. Shannon. a few parity effects concerning t-core walls; N. Robbins, M.V. Subbarao. Generalized Pell numbers and polynomials; A.G. Shannon, A.F. Horadam. an additional notice on Lucasian numbers; L. Somer. a few buildings and theorems in Goldpoint geometry; J.C. Turner. a few functions of triangle changes in Fibonacci geometry; J.C. Turner. Cryptography and Lucas series discrete logarithms; W.A. Webb. Divisibility of an F-L style convolution; M. Wiemann, C. Cooper. producing features of convolution matrices; Yongzhi (Peter) Yang. F-L illustration of department of polynomials over a hoop; Chizhong Zhou, F.T. Howard. topic Index

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First, we may try to argue like this: There are 40 students altogether in the class; take away from this the number of those having Beatles pictures (18), those having Rolling Stones picture (16), and those having Elvis pictures (12); so we take away 18 + 16 + 12. We get −6; this negative number warns us that there must be some error in our calculation; but what was not correct? We made a mistake when we subtracted the number of those students who collected the pictures of two groups twice! For example, a student having the Beatles and Elvis Presley was subtracted with the Beatles collectors as well as with the Elvis Presley collectors.

9 Prove that the sum of the first n powers of 2 (starting with 1 = 20 ) is 2n − 1. ” instead of further repetition. There is nothing wrong with this, if the argument is sufficiently simple so that we can intuitively see where the repetition leads. ” in cases where the outcome of the repetition is not so transparent. The precise way of doing this is using induction, as we are going to illustrate by revisiting some of our results. 1 (recall that the answer is 2n ). As the Principle of Induction tells us, we have to check that the assertion is true for n = 0.

Suppose that we want to prove a property of positive integers. Also suppose that we can prove two facts: (a) 1 has the property, and (b) whenever n−1 has the property, then n also has the property (n > 1). 1 Induction 27 The Principle of Induction says that if (a) and (b) are true, then every natural number has the property. This is precisely what we did above. We showed that the ”sum” of the first 1 odd numbers is 12 , and then we showed that if the sum of the first n − 1 odd numbers is (n − 1)2 , then the sum of the first n odd numbers is n2 , for whichever integer n > 1 we consider.

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