A084057-A110940, and a bunch of duplicates.

[Lambert Herrgesell]

Dear seqfans,

There are several sequences with a similar name, for example

A110940 Starting a priori with the fraction 1/1, the numerators of fractions
        built according to the rule: add top and bottom to get the new
bottom,
        add top and 5 times the bottom to get the new top.

most of them are duplicates.

instead of reading a(n) as the numerator of a fraction 1/1, read it as
separeate
sequence elements 1,1, i.e. a(n-2), a(n-1),
then (with a sign switch of a permutation),  a(n) = a(n-1)+5a(n-2) +(a(n-1)
- a(n-2)) = 2a(n-1)+4a(n-2)
(for both the top and the bottom part)


To make the dubious more obvious, the conversion by example:

a(1)=1,a(2)=1;

a(3) = a(2) + 5a(1)
b(3) = a(2) +  a(1)

a(4) = a(3) + 5b(3) // substitute b(3) by a(2) +  a(1)
     = a(3) + 5(a(2) + a(1))
     = a(3) + 5a(2) + 5a(1)  // extract another a(3)
     = 2a(3) + 4a(2)
and so on and so on ...

so a(n) = 2a(n-1) + 4a(n-2) which is A084057.


A similar argument also makes

A015519 - A110948
A001333 - A110934
A046717 - A110938
A015518 - A110939
A002533 - A110942
A002532 - A110943
A084058 - A110946 (A083100)

duplicates.


best
        Lambert
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