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Re: a[n],b[n]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104916] Re: [mg104889] a[n],b[n]
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Fri, 13 Nov 2009 05:54:28 -0500 (EST)
  • Reply-to: hanlonr at cox.net

Use RSolve

RSolve[{
   a[n + 1] ==
    1/4*(1 + Sqrt[5])*a[n] -
     1/2*Sqrt[1/2*(5 - Sqrt[5])]*b[n],
   b[n + 1] ==
    1/2*Sqrt[1/2*(5 - Sqrt[5])]*a[n] +
     1/4*(1 + Sqrt[5])*b[n],
   a[1] == 4, b[1] == 9},
  {a[n], b[n]}, n] // Simplify

{{a[n] -> (1/(1 + Sqrt[5]))*(2^(-(2*n) - 3)*
            ((4 + 9*I)*(6 + 2*Sqrt[5] - I*Sqrt[50 - 10*Sqrt[5]] - 
                    I*Sqrt[10 - 2*Sqrt[5]])*(1 + Sqrt[5] + 
                      I*Sqrt[10 - 2*Sqrt[5]])^n + (9 + 4*I)*
                 (Sqrt[50 - 10*Sqrt[5]] + Sqrt[10 - 2*Sqrt[5]] - 
                    2*I*(3 + Sqrt[5]))*(1 + Sqrt[5] - 
                      I*Sqrt[10 - 2*Sqrt[5]])^n)), 
     b[n] -> (1/(1 + Sqrt[5]))*(2^(-(2*n) - 3)*
            ((9 + 4*I)*(6 + 2*Sqrt[5] + I*Sqrt[50 - 10*Sqrt[5]] + 
                    I*Sqrt[10 - 2*Sqrt[5]])*(1 + Sqrt[5] - 
                      I*Sqrt[10 - 2*Sqrt[5]])^n - (4 + 9*I)*
                 (Sqrt[50 - 10*Sqrt[5]] + Sqrt[10 - 2*Sqrt[5]] + 
                    2*I*(3 + Sqrt[5]))*(1 + Sqrt[5] + 
                      I*Sqrt[10 - 2*Sqrt[5]])^n))}}


Bob Hanlon

---- ynb <wkfkh056 at yahoo.co.jp> wrote: 

=============
a[n + 1] = 1/4*(1 + Sqrt[5])*a[n] - 1/2*Sqrt[1/2*(5 - Sqrt[5])]*b[n],
b[n + 1] = 1/2*Sqrt[1/2*(5 - Sqrt[5])]*a[n] + 1/4*(1 + Sqrt[5])*b[n],
a[1] = 4, b[1] = 9.


a[n]=
b[n]=




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