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A080097
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a(n) = Fibonacci(n+2)^2 - 1.
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15
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0, 3, 8, 24, 63, 168, 440, 1155, 3024, 7920, 20735, 54288, 142128, 372099, 974168, 2550408, 6677055, 17480760, 45765224, 119814915, 313679520, 821223648, 2149991423, 5628750624, 14736260448, 38580030723, 101003831720
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OFFSET
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0,2
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COMMENTS
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a(n), a(n)+1 and a(n)+2 are consecutive members of A049997.
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LINKS
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FORMULA
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If n is odd, then a(n) = F(n+1)*F(n+3) = F(n)*F(n+4) - 2, else a(n) = F(n)*F(n+4) = F(n+1)*F(n+3) - 2, where F(n) = Fibonacci numbers (A000045).
a(n) = (Lucas(2*n+4) - 2*(-1)^n - 5)/5.
Sum_{n>=1} 1/a(n) = (43 - 15*sqrt(5))/18 = 29/9 - 5*phi/3, where phi is the golden ratio (A001622). - Amiram Eldar, Oct 20 2020
a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - Joerg Arndt, Nov 13 2023
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MATHEMATICA
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CoefficientList[Series[(3x+2x^2-x^3)/(1-x^2)(1-2x-2x^2+x^3)), {x, 0, 30}], x]
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PROG
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(Magma) [Fibonacci(n+2)^2 -1: n in [0..30]]; // G. C. Greubel, Jul 23 2019
(Sage) [fibonacci(n+2)^2 -1 for n in (0..30)] # G. C. Greubel, Jul 23 2019
(GAP) List([0..30], n-> Fibonacci(n+2)^2 -1); # G. C. Greubel, Jul 23 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Jan 29 2003
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EXTENSIONS
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STATUS
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approved
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