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User:FUNG Cheok Yin/proof(i) A346526

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Limit of the ratio of a(n) to a(n-1) in A346526

Statement: , where an = A346526(n) (proposed by Stefano Spezia, Jul 22 2021).

Proof

The sequence is infinite; obviously we have and .

Since , , and are definite terms in the sequence for non-negative integer ; therefore for arbitrary , we have

,

where integer can be at least one member of {1, 2, 3, 4}.

Hence we can have , where real number satisfies .

(For example:

for , we have ;
for , we have ;
for , we have ;
for , we have .)


=
=
=
= 1 + 0 + 0
= 1 .

Written on Aug 12 2021