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A121646
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a(n) = Fibonacci(n-1)^2 - Fibonacci(n)^2.
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9
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-1, 0, -3, -5, -16, -39, -105, -272, -715, -1869, -4896, -12815, -33553, -87840, -229971, -602069, -1576240, -4126647, -10803705, -28284464, -74049691, -193864605, -507544128, -1328767775, -3478759201, -9107509824, -23843770275, -62423800997, -163427632720
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OFFSET
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1,3
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COMMENTS
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Negated first differences of A007598.
Real part of (F(n-1) + i*F(n))^2. Corresponding imaginary part = A079472(n); e.g., (3 + 5i)^2 = (-16 + 30i) where 30 = A079472(5). Consider a(n) and A079472(n) as legs of a Pythagorean triangle; then hypotenuse = corresponding n-th term in the sequence (1, 2, 5, 13, ...; i.e., odd-indexed Fibonacci terms). a(n)/a(n-1) tends to Phi^2.
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REFERENCES
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Daniele Corradetti, La Metafisica del Numero, 2008
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LINKS
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FORMULA
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a(n) = Re(F(n-1) + F(n)*i)^2 = (F(n-1))^2 - (F(n))^2.
G.f.: (1-3*x)/((1+x)*(1 - 3*x + x^2)). - Paul Barry, Oct 13 2006
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EXAMPLE
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a(5) = -16 since Re(3 + 5i)^2 = (-16 + 30i).
a(5) = -16 = 3^2 - 5^2.
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MAPLE
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combinat[fibonacci](n+1)*combinat[fibonacci](n-2) ;
-% ;
end proc:
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MATHEMATICA
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f[n_] := Re[(Fibonacci[n - 1] + I*Fibonacci[n])^2]; Array[f, 29] (* Robert G. Wilson v, Aug 16 2006 *)
lst={}; Do[a1=Fibonacci[n]*Fibonacci[n+1]; a2=Fibonacci[n+1]*Fibonacci[n+2]; AppendTo[lst, 3*a1-a2], {n, 0, 60}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)
-Differences[Fibonacci[Range[0, 30]]^2] (* Harvey P. Dale, Nov 01 2022 *)
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PROG
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(Magma) [-Fibonacci(n-2)*Fibonacci(n+1): n in [1..40]]; // G. C. Greubel, Jan 07 2019
(Sage) [-fibonacci(n-2)*fibonacci(n+1) for n in (1..40)] # G. C. Greubel, Jan 07 2019
(GAP) List([1..40], n -> -Fibonacci(n-2)*Fibonacci(n+1)); # G. C. Greubel, Jan 07 2019
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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