Catalan's Identity
There are two identities known as Catalan's identity.
The first is
| F_n^2-F_(n+r)F_(n-r)=(-1)^(n-r)F_r^2, |
where F_n is a Fibonacci number. Letting r=1 gives Cassini's Identity.
The second is the trivariate identity on partition of cubes into a sum of three squares given by
| (x^2+y^2+z^2)^3=[x(=3z^2-x^2-y^2)]^2+[y(3z^2-x^2-y^2)]^2 +[z(z^2-3x^2-3y^2)]^2. |
See also
Cassini's Identity, d'Ocagne's Identity, Fibonacci NumberExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Catalan's Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CatalansIdentity.html