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Consider f(n)=3^f(n-1)-2^f(n-2) with f(1)=0 and f(2)=a.

For a>>1.51... the sequence goes to oo.

For a<<1.51... the sequence goes to 0.

Can we express the constant x=1.51... ?

Is a sine type wave possible for f(2)= x ?

What happens with complex numbers ?

Posts: 1,354

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Notice that 1.51... is NOT a real repelling fixpoint of x=3^x-2^x.

The only fixpoints of x=3^x-2^x on the extended real line are 0,1 and +oo.

The value 1 seems to suggest that f(2)=1.51... might make the sequence oscillate around 1.

Assuming the above it might work to compute 1.51... by starting at value 1 and doing the iterations backwards until we reach approximately 0. But of course nicer ways should be available.

I have not seen this sequence investigated before and it seems it might require new ideas, unless I overlooked something trivial.

I must add that this thread also deserves some credit from my student mick who some of you have already met on MSE.

Mick and I have given eachother permission to post eachothers (or common ) ideas concerning some math subjects including dynamical systems and tetration.

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I must note that - although trivial - if the sequence goes above 1 the sequence explodes to oo. This implies that solving for 1 must indeed give my desired constant.

The value of my constant is 0,516553200406323...

This value reminds me of eulers constant ?

tommy1729