Generalized Fibonacci sequence including imaginary numbers

where  is Golden ratio.

Fibonacci family

fibfam(i)=x0                                                       for i=0

fibfam(i)=x1                                                       for i=1

fibfam(i)=c_2*fibfam(i-2)+c_1*fibfam(i-1)    for i=2,3,4,. . .

 (x0,x1,c_1,c_2)=(0,1,1,1)   Fibonacci sequence

 (x0,x1,c_1,c_2)=(2,1,1,1)   Lucas number

 (x0,x1,c_1,c_2)=(0,1,2,1)   Pell number

(1)   General form of Pell number

(2)   Matrix form of Pell number

      for n=1,2,3, . . .