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A121646 a(n) = Fibonacci(n-1)^2 - Fibonacci(n)^2. 8
-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; text; internal format)
OFFSET

1,3

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

REFERENCES

Daniele Corradetti, La Metafisica del Numero, 2008

LINKS

Table of n, a(n) for n=1..29.

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

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

EXAMPLE

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

a(5) = -16 = 3^2 - 5^2.

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]*Fibonacci[n+3], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

PROG

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

CROSSREFS

Cf. A079472.

Sequence in context: A077551 A106588 A123785 * A226205 A300533 A221783

Adjacent sequences:  A121643 A121644 A121645 * A121647 A121648 A121649

KEYWORD

sign,easy

AUTHOR

Gary W. Adamson, Aug 13 2006

STATUS

approved

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Last modified October 22 14:49 EDT 2018. Contains 316488 sequences. (Running on oeis4.)