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A244855
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a(n) = Fibonacci(n)^4-1.
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4
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0, 0, 15, 80, 624, 4095, 28560, 194480, 1336335, 9150624, 62742240, 429981695, 2947295520, 20200652640, 138458409999, 949005240560, 6504586067280, 44583076827135, 305577005139120, 2094455819300624, 14355614096087055, 98394841894789440, 674408281676875200
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OFFSET
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1,3
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COMMENTS
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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).
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LINKS
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Hideyuki Ohtsuka, Problem B-1174, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 53, No. 3 (2015), p. 273; A Summation of Recipricals, Solution to Problem B-1174 by Steve Edwards, ibid., Vol. 54, No. 3 (2016), pp. 277-278.
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FORMULA
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G.f.: x^3*(-15-5*x+x^2) / ( (x-1)*(x^2-7*x+1)*(x^2+3*x+1) ).
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
Sum_{n>=3} (-1)^(n+1)/a(n) = 1/18 (Ohtsuka, 2015). - Amiram Eldar, Dec 09 2021
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EXAMPLE
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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.
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MAPLE
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with(numtheory):with(combinat, fibonacci):nn:=30:for i from 1 to nn do:x:=fibonacci(i)^4-1: printf(`%d, `, x):od:
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MATHEMATICA
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LinearRecurrence[{5, 15, -15, -5, 1}, {0, 0, 15, 80, 624}, 30] (* Harvey P. Dale, Dec 01 2019 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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