A087425 a(n) = S(5*n,5)/S(n,5) where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).
23105, 459119455, 9758296035305, 208416652653910655, 4452963734477926435505, 95143212432467064852443605, 2032859482921447476046969568705, 43434715031065603778576465510557055
Offset: 1
Keywords
Programs
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Maple
S:=proc(n,m) add(binomial(n,k)*combinat:-fibonacci(m*k),k=0..n) end: m:=5: seq(S(m*n,m)/S(n,m),n=1..16); # Georg Fischer, Jul 07 2021
Formula
a(n) = 121^n+{(21367+9555*sqrt(5))/2}^n+{(21367-9555*sqrt(5))/2}^n+{(1617+715*sqrt(5))/2}^n+{(1617-715*sqrt(5))/2}^n.
a(n) = (x_1)^n+(x_2)^n+(x_3)^n+(x_4)^n+(x_5)^n where (x_i) (1<=i<=5) are the roots of X^5-23105*X^4+37360785*X^3-4520654985*X^2+40931916905*X-25937424601.