Os numeros mágicos de Ball e a sequência de Fibonacci

Abstract

Let $ x_n $ be a number with $ n $ digits. For $ n \geq 2 $, the number of $ n $ digits obtained by reversing the position of the digits of $ x_n $ is called the reverse number of $ x_n $ and is indicated by $ x_n '$. Admit $ x_n> x_n '$ and write the magic number of Ball $ B = (x_n - x_n') + (x_n - x_n ')' $. In Webster \cite{webs}, and independently in Costa e Mesquita \cite{costa2}, it is shown that every Ball's number $ B $ is a multiple of 99. For each entire $ k \geq 0 $, consider $ x_ {2k} $ (or $ x_ {2k + 1} $) any number and $ B (k) $ associates the number of possible magic numbers for Ball, that is, corresponding to the numbers of digits $ 2, 4, 6, \ldots, 2k, \ldots $ (or $ 3, 5, 7, \ldots, 2k + 1, \ldots $) we have a string associated with $ 1, 4, 12, \ldots, B (k), \ldots $ representing Ball's number of amount. Webster considers the particular case in which the first digit of the number $ x_ {n} $ is always greater than the last digit and shows that $ B (k) $ is a term in the Fibonacci sequence. While in Costa e Mesquita it is shown that for every integer $ k \geq 2 $ the amount $ B (k) $ is between $ B(k-1) $ and $ 3^{k-1} + B (k-1 ) $. Here we are going to improve the Webster result and show that the quantity $ B (k) $ is the sum of two terms of the Fibonacci sequence.