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A121646
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Fibonacci(n-1)^2 - Fibonacci(n)^2.
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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| 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.
3*A001654(n)-A001654(n+1)=A121646(n). [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009]
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REFERENCES
| Daniele Corradetti, La Metafisica del Numero, 2008
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (2,2,-1).
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FORMULA
| 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 (pbarry(AT)wit.ie), Oct 13 2006
a(n)= -F(n+1)*F(n-2) where F=A000045 [From Ron Knott (enquiry(AT)ronknott.com), Jan 24 2009]
a(n) = (4*(-1)^n-|A098149(n)|)/5 . - R. J. Mathar, Jan 13 2011
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EXAMPLE
| a(5) = -16 since Re:(3 + 5i)^2 = (-16 + 30i).
a(5) = -16 = 3^2 - 5^2.
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MAPLE
| seq(-1*(fibonacci(i)*fibonacci(i+3)), i=-1..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 23 2007
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MATHEMATICA
| 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 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009]
Table[Fibonacci[n]*Fibonacci[n+3], {n, 0, 40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009]
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CROSSREFS
| Cf. A079472.
Sequence in context: A077551 A106588 A123785 * A099101 A038120 A192911
Adjacent sequences: A121643 A121644 A121645 * A121647 A121648 A121649
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KEYWORD
| sign
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 13 2006
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EXTENSIONS
| More terms from Robert G. Wilson v, Aug 16 2006
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