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Old 2021-02-07, 15:55   #13
NHoodMath
 
Jan 2017

22·7 Posts
Default Other closed form of the Hugo sequence and observations

There is a much simpler closed form for the Hugo Numbers than what has been posted by Hugo and the others, they are equal to:

(Sum from k=0 to k=(p-1)/2 of (Binomial(p, 2*k)*13^k))/2^(n-1)

It can also be shown that the Hugo Numbers are, in the limit, a geometric series, with first term 1/2(1+sqrt(13)) and common ratio 1/2(7+sqrt(13)).

There are also Mersenne-like primality characteristics of this sequence that I have observed (Hugo(y*x) = Hugo(y) * Hugo(x) * cofactor unless y*x is a perfect power, divisors of Hugo(n) with N prime of specific forms either 2*k*p+1 or 2*k*p-1)
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