On Wed, May 13, 2020 at 2:32 PM <rcs@xmission.com> wrote:
(from Bernie Cosell <bernie@fantasyfarm.com>) (The list filtering software intercepted this message, without telling me why. I've retyped the formulas. --rcs)
Binet’s formula for the Fibonacci numbers follows fairly easily from the assumption that F_n=x^n, but I didn’t quite follow how/why one would make that guess. I tried to see if I could come up with Binet’s formula not “guessing” that form. Instead I tried a finite polynomial F_n = sum(i=0,n,a_i*x^i). But then I’m stuck. I’d love to have discovered that this implies a_i=0 for all i/=n and a_i=1 for i=n and then Binet’s formula follows.
But it doesn't imply that; a_i = 0 for all except i = 7 is a solution, too. Or any number of the a_i can be nonzero, as long as each of the associated x_u are either phi or phi-bar (other solution of the same quadratic). Solving something with multiple solutions is going to be harder than solving something with 1 solution. Since the constraint F_n = F_(n-1) + F_(n-2) is linear, than given any two series that are solutions, their sum (and in fact any linear combination of them) is a solution too. So looking for a solution that's a sum of a bunch of terms is going to be hugely underdetermined. Better first to find solutions that are a single summand, and then if you've found all of those, then all the things you can make as linear combinations of those will also be a solution. Then you can use the initial conditions F(0) = F(1) = 1 to determine which linear combination you want. Is this helpful? Andy
But I can’t quite figure that out. When I take that summation and plug it into F_n = F_(n-1) + F_(n-2) the best I can come up with a_n x^n = F_(n-2), which doesn’t look promising. What am I missing?
I know this isn't much "fun" but for odd reasons I've been revisiting math stuff that seemed obvious to me 50+ years, but has lain dormant. Thanks /bernie\
Bernie Cosell bernie@fantasyfarm.com -- Too many people; too few sheep --
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com