Morgado Identity

Author

Eric W. Weisstein
November 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/MorgadoIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MorgadoIdentity.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Check

(Fibonacci[n]) (Fibonacci[n+1]) (Fibonacci[n+2]) (Fibonacci[n+4]) (Fibonacci[n+5]) (Fibonacci[n+6])+((((LucasL[n+3])))^2)==((((Fibonacci[n+3]) (2 (Fibonacci[n+2]) (Fibonacci[n+4])-((((Fibonacci[n+3])))^2))))^2)

FullSimplify[Subtract@@{Times@@(Fibonacci[n+#]&/@{0,1,2,4,5,6})+LucasL[n+3]^2,
(Fibonacci[n+3](2Fibonacci[n+2]Fibonacci[n+4]-Fibonacci[n+3]^2))^2}]

FullSimplify[FunctionExpand[%],n∈Integers&&n>0]

Table[Times@@(Fibonacci[n+#]&/@{0,1,2,4,5,6})+LucasL[n+3]^2-(Fibonacci[n+3](2Fibonacci[n+2]Fibonacci[n+4]-Fibonacci[n+3]^2))^2,{n,20}]//Simplify