[seqfan] maybe not surprising.

[Wouter Meeussen]

hi,

maybe the following is trivial:

In[  ]:= try1= Sum[   Binomial[n+k ,k-n]  ,{n,0,k}]
Out[  ]= (2*Cosh[(-1 - 2*k)*ArcSinh[1/2]])/Sqrt[5]

generates the odd terms from the Fibonacci series.

some different ways to write the same:

(2*Cosh[ArcCsch[2] + 2*k*ArcCsch[2]])/Sqrt[5]
==
((1/2 + Sqrt[5]/2)^(-1 - 2*k) + (1/2 + Sqrt[5]/2)^(1 + 2*k))/Sqrt[5]
==
((2/(1 + Sqrt[5]))^(1 + 2*k) + (1/2*(1 + Sqrt[5]))^(1 + 2*k))/Sqrt[5]

wouter.

Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
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