Proof: In[302]:= Table[ChebyshevU[n, I/2]/I^n, {n, -1, 9}] Out[302]= {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55} In[303]:= Sum[ChebyshevU[n, I/2] t^(n + 1)/I^n, {n, -1, ∞}] Out[303]= (t (-Sqrt[2 (3 + Sqrt[5])] + Sqrt[10 (3 + Sqrt[5])] + 4 t + Sqrt[2 (3 + Sqrt[5])] t^2 - Sqrt[10 (3 + Sqrt[5])] t^2))/(4 (1 - 3 t^2 + t^4)) In[304]:= FullSimplify@% Out[304]= -(t/(-1 + t + t^2)) In[305]:= Series[%,{t,0,9}] Out[305]= t+t^2+2 t^3+3 t^4+5 t^5+8 t^6+13 t^7+21 t^8+34 t^9+O[t]^10 —rwg On Thu, Mar 19, 2020 at 7:06 PM Bill Gosper <billgosper@gmail.com> wrote:
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}