Abstract: We exhibit a family of sequences of noncommutative variables, recursively defined using monic palindromic polynomials in Q[x], and show that each possesses the Laurent phenomenon. This generalizes a conjecture by Kontsevich.

Abstract: We exhibit a family of sequences of noncommutative variables, recursively defined using monic palindromic polynomials in Q[x], and show that each possesses the Laurent phenomenon. This generalizes a conjecture by Kontsevich.

Abstract: The study of sequences extends in a wide range from the golden ratio to the current financial trading algorithm. Although not realized by many, both the Fibonacci and Lucas sequences are incorporated into the algorithm today. In addition, among the far-reaching applications of these numbers are the Fibonacci search method and the Fibonacci heap data structure in computer science. Hence, mathematicians seek to find more results and further studies into these sequences for the purpose of greater potential benefits. In the area of mathematics, Fibonacci and Lucas numbers are used in connection to efficient primality testing of Mersenne numbers; the method revolves around the fact that if F(n) is prime, then n is prime with the exception for F(4)=3. Some of the largest known primes were discovered via this process. Yet, despite the extensive literature on Fibonacci and Lucas numbers, there are still many questions, which are still unanswered. Recently, several authors considered the problem of estimating expressions of the classical Fibonacci and Lucas sequences given by F(0) = 0, F(1) =1,F(n) =F(n-1)+F(n-2) and L0 =2,L1 =1,Ln =L(n-1)+L(n-2), respectively. In this paper, we find the exact asymptotic behavior for a class of infinite partial sums whose general term is a negative integer power of L(k). Most of the previous works focused on the relatively simple cases s = 1 and s = 2 where s represents the power. For this paper, we find the general idea in which expressions could be found for cases of s from 1 to 6.

Abstract: The study of sequences extends in a wide range from the golden ratio to the current financial trading algorithm. Although not realized by many, both the Fibonacci and Lucas sequences are incorporated into the algorithm today. In addition, among the far-reaching applications of these numbers are the Fibonacci search method and the Fibonacci heap data structure in computer science. Hence, mathematicians seek to find more results and further studies into these sequences for the purpose of greater potential benefits. In the area of mathematics, Fibonacci and Lucas numbers are used in connection to efficient primality testing of Mersenne numbers; the method revolves around the fact that if F(n) is prime, then n is prime with the exception for F(4)=3. Some of the largest known primes were discovered via this process. Yet, despite the extensive literature on Fibonacci and Lucas numbers, there are still many questions, which are still unanswered. Recently, several authors considered the problem of estimating expressions of the classical Fibonacci and Lucas sequences given by F(0) = 0, F(1) =1,F(n) =F(n-1)+F(n-2) and L0 =2,L1 =1,Ln =L(n-1)+L(n-2), respectively. In this paper, we find the exact asymptotic behavior for a class of infinite partial sums whose general term is a negative integer power of L(k). Most of the previous works focused on the relatively simple cases s = 1 and s = 2 where s represents the power. For this paper, we find the general idea in which expressions could be found for cases of s from 1 to 6.

Nathaniel Shar, Rutgers Experimental Mathematics Seminar, October 24, 2013.

Abstract: If a(n) is the sum of the cubes of the entries on the nth row of Pascal's triangle, then (n+1)^2 a(n) = (7n^2 - 7n + 2)a(n-1) + 8(n-1)^2a(n-2). It seems challenging for a human to find a bijective proof of this, but a computer can do it, with a little help. I'll show you a real live bijection, implemented of course by the computer, that proves this identity, and describe a method that might help computers bijectify other difficult identities.

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