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A244855 a(n) = Fibonacci(n)^4-1. 2
0, 0, 15, 80, 624, 4095, 28560, 194480, 1336335, 9150624, 62742240, 429981695, 2947295520, 20200652640, 138458409999, 949005240560, 6504586067280, 44583076827135, 305577005139120, 2094455819300624, 14355614096087055, 98394841894789440, 674408281676875200 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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).

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 ;

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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

FORMULA

From R. J. Mathar, Nov 02 2014: (Start)

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

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

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

Table[(Fibonacci[n]^4 - 1), {n, 40}] (* Vincenzo Librandi, Jul 26 2014 *)

LinearRecurrence[{5, 15, -15, -5, 1}, {0, 0, 15, 80, 624}, 30] (* Harvey P. Dale, Dec 01 2019 *)

PROG

(MAGMA) [Fibonacci(n)^4-1: n in [1..30]]; // Vincenzo Librandi, Jul 26 2014

(PARI) a(n) = fibonacci(n)^4-1; \\ Michel Marcus, Oct 20 2020

CROSSREFS

Cf. A000045, A001622, A056571, .

Sequence in context: A050149 A055815 A338414 * A102360 A309336 A266288

Adjacent sequences:  A244852 A244853 A244854 * A244856 A244857 A244858

KEYWORD

nonn,easy

AUTHOR

Michel Lagneau, Jul 25 2014

STATUS

approved

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Last modified April 22 16:06 EDT 2021. Contains 343177 sequences. (Running on oeis4.)