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

-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

Offset: 1

Author

Gary W. Adamson, Aug 13 2006

Keywords

• sign
• easy

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). - Vladimir Joseph Stephan Orlovsky, Nov 17 2009

Examples

			a(5) = -16 since Re(3 + 5i)^2 = (-16 + 30i).
a(5) = -16 = 3^2 - 5^2.
		

References

• Daniele Corradetti, La Metafisica del Numero, 2008.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A079472.

Programs

GAP

List([1..40], n -> -Fibonacci(n-2)*Fibonacci(n+1)); # G. C. Greubel, Jan 07 2019

Magma

[-Fibonacci(n-2)*Fibonacci(n+1): n in [1..40]]; // G. C. Greubel, Jan 07 2019

Maple

A121646 := proc(n)
    combinat[fibonacci](n+1)*combinat[fibonacci](n-2) ;
    -% ;
end proc:
seq(A121646(n),n=1..10) ; # R. J. Mathar, Jun 22 2017

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 (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)
Table[-Fibonacci[n-2]*Fibonacci[n+1],{n,1,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)
-Differences[Fibonacci[Range[0,30]]^2] (* Harvey P. Dale, Nov 01 2022 *)

PARI

a(n) = fibonacci(n-1)^2 - fibonacci(n)^2 \\ Charles R Greathouse IV, Jun 11 2015

Sage

[-fibonacci(n-2)*fibonacci(n+1) for n in (1..40)] # G. C. Greubel, Jan 07 2019

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, Oct 13 2006

a(n) = -F(n+1)*F(n-2) where F=A000045. - Ron Knott, Jan 24 2009

a(n) = (4*(-1)^n - |A098149(n)|)/5. - R. J. Mathar, Jan 13 2011