Bill, Some claim Dijkstra's recursion: http://www.cs.utexas.edu/users/EWD/ewd06xx/EWD654.PDF is very fast at calculating F_n, however it seems that using F_{a+b} = F_{a+1} F_b + F_a F_{b-1} recursively and splitting n = a+b such that a = 2^k leads to caching just triplets around each 2^k pivot (k greatest such that 2^k < n). This caching is harder to do as effectively with Dijkstra's algorithm, so the splitting F_{a+b} method becomes competitive while using less memory. Do you have an opinion on the subject? Andres. On 9/19/18 20:11 , Bill Gosper wrote:

World's smallest(?) fibonacci formula: F_n+1 = U_n(i/2)/i^n. where U:= Чебышёв polynomial, 2nd kind := sin((n+1)acos z)/√(1-z)/√(1+z). This provides an alternate (complex) interpolation for nonintegers. Slightly nonobvious: With this definition, Im(F_x+1)/Im(F_x) = -1/GoldenRatio (x real noninteger), which can be coaxed out of Mathematica. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun