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A121646 a(n) = Fibonacci(n-1)^2 - Fibonacci(n)^2.

%I A121646 #38 Feb 06 2025 09:05:15
%S A121646 -1,0,-3,-5,-16,-39,-105,-272,-715,-1869,-4896,-12815,-33553,-87840,
%T A121646 -229971,-602069,-1576240,-4126647,-10803705,-28284464,-74049691,
%U A121646 -193864605,-507544128,-1328767775,-3478759201,-9107509824,-23843770275,-62423800997,-163427632720
%N A121646 a(n) = Fibonacci(n-1)^2 - Fibonacci(n)^2.
%C A121646 Negated first differences of A007598.
%C A121646 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.
%C A121646 3*A001654(n) - A001654(n+1) = A121646(n). - _Vladimir Joseph Stephan Orlovsky_, Nov 17 2009
%D A121646 Daniele Corradetti, La Metafisica del Numero, 2008.
%H A121646 G. C. Greubel, <a href="/A121646/b121646.txt">Table of n, a(n) for n = 1..1000</a>
%H A121646 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A121646 a(n) = Re(F(n-1) + F(n)*i)^2 = (F(n-1))^2 - (F(n))^2.
%F A121646 G.f.: (1-3*x)/((1+x)*(1 - 3*x + x^2)). - _Paul Barry_, Oct 13 2006
%F A121646 a(n) = -F(n+1)*F(n-2) where F=A000045. - _Ron Knott_, Jan 24 2009
%F A121646 a(n) = (4*(-1)^n - |A098149(n)|)/5. - _R. J. Mathar_, Jan 13 2011
%e A121646 a(5) = -16 since Re(3 + 5i)^2 = (-16 + 30i).
%e A121646 a(5) = -16 = 3^2 - 5^2.
%p A121646 A121646 := proc(n)
%p A121646     combinat[fibonacci](n+1)*combinat[fibonacci](n-2) ;
%p A121646     -% ;
%p A121646 end proc:
%p A121646 seq(A121646(n),n=1..10) ; # _R. J. Mathar_, Jun 22 2017
%t A121646 f[n_] := Re[(Fibonacci[n - 1] + I*Fibonacci[n])^2]; Array[f, 29] (* _Robert G. Wilson v_, Aug 16 2006 *)
%t A121646 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 *)
%t A121646 Table[-Fibonacci[n-2]*Fibonacci[n+1],{n,1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 17 2009 *)
%t A121646 -Differences[Fibonacci[Range[0,30]]^2] (* _Harvey P. Dale_, Nov 01 2022 *)
%o A121646 (PARI) a(n) = fibonacci(n-1)^2 - fibonacci(n)^2 \\ _Charles R Greathouse IV_, Jun 11 2015
%o A121646 (Magma) [-Fibonacci(n-2)*Fibonacci(n+1): n in [1..40]]; // _G. C. Greubel_, Jan 07 2019
%o A121646 (Sage) [-fibonacci(n-2)*fibonacci(n+1) for n in (1..40)] # _G. C. Greubel_, Jan 07 2019
%o A121646 (GAP) List([1..40], n -> -Fibonacci(n-2)*Fibonacci(n+1)); # _G. C. Greubel_, Jan 07 2019
%Y A121646 Cf. A079472.
%K A121646 sign,easy
%O A121646 1,3
%A A121646 _Gary W. Adamson_, Aug 13 2006