# Lucas Numbers and Generating Functions

by Johnny Numbers
Tags: functions, generating, lucas, numbers
 P: 9 Here is my problem and my attempt at the answer. Any help or advice is highly appreciated. Problem With the famous sequence of Lucas numbers 1, 3, 4, 7, 11, 18... (Where each number is the sum of the last two and the first two are defined as 1 and 3.) use generating functions to find an explicit formula for the Lucas function. Attempted Solution We have $$\sum_{j=1}^{n}F_jx^j$$ where Fj denotes the jth Fibonacci number and n is going to infinity. Then we add that to $$\sum_{j=-1}^{n}F_jx^j^+^2$$ Where F-1 = -1 and F0 = 0 And that should get us a function of Lucas numbers right?
 Sci Advisor HW Helper P: 3,144 The Lucas numbers satisfy the relation $$L_{n+1} = L_n + L_{n-1}$$ Just set $L_n = a^n$ and solve for a. Your generating function will be a linear combination of the two solutions. Apply your initial conditions ( $L_1$ and $L_2$) to determine the two arbitrary constants and you're done! :)
 Sci Advisor HW Helper P: 3,144 Lucas Numbers and Generating Functions In that case, it should be apparent that $L_n = F_{n+1} + F_{n-1}$