let @=3+sqrt(5)
so @^n = (3+sqrt(5))^n =(an+bn*sqrt(5))
@^(n-1)=(3+sqrt(5))*(an+bn*sqrt(5))
so an+1=3an+5bn
bn+1=an+3bn
so [an bn]= A [an-1 bn -1]
= A^n [a0 b0] where a0=1 and b0=0
where a= [ 3 5,1 3]
calculate an and bn in each step and devide it by 1000 for next an and bn
at end step ans=(an + bn*sqrt(5)) mod 1000
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