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The d'Ocagne's identity is normally stated as $(-1)^n F_{m-n} = F_m F_{n+1}-F_n F_{m+1}$.

Every book about the Fibonacci numbers has this formula in it, but I can't find any context about it. When it was first published, why it deserves a name or even if originally $m,n$ were considered to be any integers.

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A special case for $m=n-1$ was discovered by Cassini in 1680, it was rediscovered by Simson in 1753. Cassini's identity $(-1)^{n+1} = F_{n-1} F_{n+1}-F_n^2$ was generalized in two ways: one was found by Catalan in 1879, and reads $(-1)^{n+k} = F_{n-k} F_{n+k}-F_n^2$, the other by d'Ocagne (1862-1938). It is unclear when exactly, but in 1889 he studied recurrence relations of Fibonacci and Lucas type, in connection with periodic continued fractions.

As for context, both identities are examples of determinantal identities for Fibonacci numbers, and show close relation between recurrence relations and matrix algebra. Koshy's survey book Fibonacci and Lucas Numbers with Applications has a chapter on determinantal identities. Cassini's identity can be used to generate Curry's paradox invented by Curry in 1953, a dissection of $8×8$ square "rearranged" into a $5×13$ rectangle despite $64\neq65$. Other cases of d'Ocagne's identity can be used to generate similar paradoxes. Recently, Spivey used determinants to derive an identity that generalizes both d'Ocagne's and Catalan's cases.

Interestingly, d'Ocagne's Wikipedia entry does not even mention this identity. He served as the president of the Mathematical Society of France in 1901 and the chair of geometry at École Polytechnique since 1912. But he is most remembered for inventing nomography, a method of graphic calculation popular with engineers before the onset of pocket calculators.

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