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A244855 a(n) = Fibonacci(n)^4-1. 2

%I

%S 0,0,15,80,624,4095,28560,194480,1336335,9150624,62742240,429981695,

%T 2947295520,20200652640,138458409999,949005240560,6504586067280,

%U 44583076827135,305577005139120,2094455819300624,14355614096087055,98394841894789440,674408281676875200

%N a(n) = Fibonacci(n)^4-1.

%C For n > 1, a(n) = F(n-2)*F(n-1)*F(n+1)*F(n+2) with the property F(n-2)+F(n-1)+ F(n+1)+ F(n+2) = F(n) + F(n+3) = 2*F(n+2).

%C F(n)^2 - 1 = F(n-1)*F(n+1) if n odd, and F(n)^2 - 1 = F(n-2)*F(n+2)if n even ;

%C F(n)^2 + 1 = F(n-2)*F(n+2) if n odd, and F(n)^2 + 1 = F(n-1)*F(n+1) if n even, hence the product (F(n)^2 - 1)*(F(n)^2 + 1)= F(n-2)*F(n-1)*F(n+1)*F(n+2).

%H Vincenzo Librandi, <a href="/A244855/b244855.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).

%F From _R. J. Mathar_, Nov 02 2014: (Start)

%F G.f.: x^3*(-15-5*x+x^2) / ( (x-1)*(x^2-7*x+1)*(x^2+3*x+1) ).

%F a(n) = A056571(n)-1. (End)

%F Sum_{n>=3} 1/a(n) = 35/18 - 5*sqrt(5)/6 = 25/9 - 5*phi/3, where phi is the golden ratio (A001622). - _Amiram Eldar_, Oct 20 2020

%e a(5) = Fibonacci(5)^4-1 = 624 = 3*8*2*13 because Fibonacci(5)^2-1=3*8 and Fibonacci(5)^2+1 = 2*13.

%p with(numtheory):with(combinat, fibonacci):nn:=30:for i from 1 to nn do:x:=fibonacci(i)^4-1: printf(`%d, `, x):od:

%t Table[(Fibonacci[n]^4 - 1), {n, 40}] (* _Vincenzo Librandi_, Jul 26 2014 *)

%t LinearRecurrence[{5,15,-15,-5,1},{0,0,15,80,624},30] (* _Harvey P. Dale_, Dec 01 2019 *)

%o (MAGMA) [Fibonacci(n)^4-1: n in [1..30]]; // _Vincenzo Librandi_, Jul 26 2014

%o (PARI) a(n) = fibonacci(n)^4-1; \\ _Michel Marcus_, Oct 20 2020

%Y Cf. A000045, A001622, A056571, .

%K nonn,easy

%O 1,3

%A _Michel Lagneau_, Jul 25 2014

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Last modified May 12 03:56 EDT 2021. Contains 343810 sequences. (Running on oeis4.)