On the relationship between the recurrences in nilpotent groups and the binomial formula (Q1769203)

If \(G\) is a nilpotent group of exponent \(p\) (\(p\) is a prime), nilpotency class \(n\) with a presentation \(G= \langle x_1,x_2,\dots, x_n,x_{n+1}| [x_2,x_1]=x_3, [x_3,x_1]=x_4,\dots,[x_n,x_1]=x_{n+1}\rangle \) then the authors examine the form of the entries of the two-step Fibonacci sequences formed by two elements of \(G\). Then they examine the relationship between the number of recurrence sums involved in the \(j\)th term of the last component of the Fibonacci sequences and the coefficient of the binomial formula \((a+b)^{n-2}, n \geq 2\).