19 Mar
2020
19 Mar
'20
8:06 p.m.
Even shorter: Fib(v+1) = |Uᵥ(*i*/2)|. In[301]:= Table[Abs@ChebyshevU[n, I/2], {n, -1, 9}] Out[301]= {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55} —rwg On Mon, Jan 7, 2019 at 6:19 AM Bill Gosper <billgosper@gmail.com> wrote:
U_n(i/2)/iⁿ, where U is the special function known as Chebyshev polynomial of the second kind:
In[964]:= Table[ChebyshevU[n, I/2]/I^n, {n, 0, 9}]
Out[964]= {1, 1, 2, 3, 5, 8, 13, 21, 34, 55} --Bill U_n(x):= In[173]:= Sin[(n + 1) ArcCos@x]/Sin@ArcCos@x
Out[173]= Sin[(1 + n) ArcCos[x]]/Sqrt[1 - x^2]
In[174]:= FullSimplify@Table[%, {n, 0, 5}]
Out[174]= {1, 2 x, -1 + 4 x^2, -4 x + 8 x^3, 1 - 12 x^2 + 16 x^4, 6 x - 32 x^3 + 32 x^5}