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Tribonacci interpolation ? - tommy1729 - 09/08/2014
I was thinking about (real) interpolation of the tribonacci sequence. Of course there is the real part of the binet like analogue. But I do not want that solution or at least I want to arrive at it in a different way. Instead of considering asymptotics , positive derivatives , fake function theory and the recent alike stuff , I was more intrested in using more "classical" stuff. Considering that a recursion is close to an iteration I got the idea to use superfunctions. --- I must note however that a continuum sum might also be usefull here !! --- But then I encountered a problem. It can probably be fixed though. This is the idea I had and the trouble I encountered ; trib(x) = trib(x-1) + trib(x-2) + trib(x-3) Define rat(x) = trib(x+1)/trib(x). Now rat(x+1) = ( trib(x+1) + trib(x) + trib(x-1) ) / trib(x+1) = 1 + (trib(x) + trib(x-1))/trib(x+1) = 1 + 1/rat(x-1) + rat(x-2)/rat(x-1) This looks familiar ... Somos , Fibonacci ... hmm. ( still thinking , might edit ) --- Notice the analogue for Fibonacci works : we get ratfibo(x) = 1 + 1/ratfibo(x) which has the golden mean and 1 - golden mean as its fixpoints. Im trying to get the tribonacci constant as a fixpoint here ... --- regards tommy1729 |