Horadam Sequences

A Horadam sequence is a generalization of the Fibonacci numbers defined by the four constants (\(p,q,r,s\)) and the definitions \(H_0=p\) and \(H_1=q\) together with the linear recurrence equation for \(n>1\):$$H_n=sH_{n-1}+rH_{n-2}$$Specific values of \(r\) and \(s\) lead to well known sequences:

• Fibonacci Numbers: \(F_n=F_{n-1}+F_{n-2}\\ \text{ where }n \geq 2, F_0=0, F_1=1\)
   
• Lucus Number: \(L_n=L_{n-1}+L_{n-2} \\ \text{ where } n \geq 2, L_0=2, L_1=1\)
  
  
• Pell Numbers: \(P_n=2P_{n-1}+P_{n-2} \\ \text{ where } n \geq 2, P_0=0, P_1=1\)
  
  
• Pell-Lucus Numbers: \(Q_n=2Q_{n-1}+Q_{n-2} \\ \text{ where } n \geq 2, Q_0=Q_1=1\)
  
  
• Jacobsthal Numbers: \(J_n=J_{n-1}+2J_{n-2} \\ \text{ where } n \geq 2, J_0=0, J_1=1 \)
  
  
• Jacobsthal-Lucas Numbers: \( j_n=j_{n-1}+2j_{n-2} \\ \text{ where } n \geq 2, j_0=j_1=2\)

Today I turned 28160 days old and this number is a member of OEIS A085449:

 A085449: Horadam sequence (0,1,4,2)

The numbers indicate that the sequence is generated as follows for \(n>1\):$$H_n=2H_{n-1}+4H_{n-2}$$with \(H_0=0\), \(H_1=1\), \(r=4\) and \(s=2\)

The sequence begins: 0, 1, 2, 8, 24, 80, 256, 832, 2688, 8704, 28160

The generating function is:$$ \frac{x}{1-2x-4x^2}$$The progressive ratios between successive terms approach the following number:$$ \frac{H_n}{H_{n-1}} \rightarrow 2\phi = \sqrt{5}+ 1 \text{ as }n \text{ gets larger}$$In the general case we have:$$ \frac{H_n}{H_{n-1}} \rightarrow \frac{r}{s} \phi = \frac{r}{s} (\sqrt{5}+ 1) \text{ as }n \text{ gets larger}$$