OFFSET
0,2
COMMENTS
C(n) = (F(n) + F(n-1)*C(0))/(F(n+1) + F(n)*C(0)) = (3*F(n)*F(n+1) + (-1)^n*(1+I))/(3*F(n)*F(n+2) + (-1)^n).
a(n) = F(n+2)^2 + F(n)^2 is the square of the short sides L(n) of the parallelogram appearing in the dissection fallacy of the square F(n+3) X F(n+3), where F(n) is the Fibonacci number A000045(n). For n >= 0, floor(L(n)/h(n)) = A014742(n) (see the proof there), where h(n) is the perpendicular distance between the long sides LL(n) = L(n+1). a(n) is also the first difference of A014742(n). See the link with an illustration. - Kival Ngaokrajang, Jun 27 2014, edited by Wolfdieter Lang, Jul 16 2014
Re(C(n)) = A014742(n)/a(n), n >= 0. For Im(C(n)) see the name. - Wolfdieter Lang, Jul 16 2014
For n>=2 a(n) is half the number of matchings in the graph consisting of two (n+1)-cycles connected by a bridge. - Birhanu Gebrehanna Habtemariam, Feb 14 2026
LINKS
Kival Ngaokrajang, Illustration of initial terms
Eric Weisstein's World of Mathematics, Dissection Fallacy
Wikipedia, Missing square puzzle
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
FORMULA
a(n) = 3*F(n)*F(n+2) + (-1)^n = 3*A059929(n) +(-1)^n, where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = ceiling(3/5*(g/2)^(n+1))-(1+(-1)^n)/2, with g = 3 + sqrt(5).
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3). - Vladeta Jovovic, May 06 2002
G.f.: (1+3*x-2*x^2)/((1+x)*(1-3*x+x^2)). - Vladeta Jovovic, May 06 2002
a(n) = F(n)^2 + F(n+2)^2. - Ron Knott, Aug 02 2004
a(n) = 2*F(n)*F(n+2) + F(n+1)^2 = F(n+1)*F(n+3) + F(n)^2 +(-1)^(n-1). - J. M. Bergot, Sep 15 2012
Equals the logarithmic derivative of A224415. - Paul D. Hanna, Apr 05 2013
2*a(n) = Fibonacci(n+1)^2 + Lucas(n+1)^2. - Bruno Berselli, Sep 26 2017
MATHEMATICA
a[n_] := 3Fibonacci[n]Fibonacci[n+2]+(-1)^n
(*A000045*) F[n_] := (((1 + Sqrt[5])/2)^n - ((1 - Sqrt[5])/2)^n)/Sqrt[5]; (*A000032*) L[n_] := ((1 + Sqrt[5])/2)^n + ((1 - Sqrt[5])/2)^n; Table[FullSimplify[ExpandAll[(F[n]^2 + L[n]^2)/2]], {n, 0, 50}] (* Roger L. Bagula, Nov 17 2008 *)
LinearRecurrence[{2, 2, -1}, {1, 5, 10}, 70] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)
PROG
(PARI) a(n)=([0, 1, 0; 0, 0, 1; -1, 2, 2]^n*[1; 5; 10])[1, 1] \\ Charles R Greathouse IV, Sep 23 2015
(Magma) [3*Fibonacci(n)*Fibonacci(n+2)+(-1)^n: n in [0..40]]; // Vincenzo Librandi, Sep 24 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, May 05 2002
EXTENSIONS
Edited by Dean Hickerson, May 08 2002
STATUS
approved
