My favorite integer sequence

[tommy1729]

I realised that solving the equation

f ' (2x) = 3 f(x)

can be done in the same way as the equation f ' (2x) = f(x) or equivalent f ' (x) = f(x/2).

Maybe that leads to a closed form solution too.
( closed form here means expressible as standard functions , sums and integrals )

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As for the equations

f(x) - f(x-1) = f(x/2)
or
f(x+1) - f(x) = f(x/2)

and Jay's comments ; Im aware of the weaknesses of the equations.

I was just trying to find a better approximation then the equation f ' (x) = f(x/2) gives.

Let A(x) be one of the Original sequences and let T(x) be the nonconstant analytic solution to T(x) - T(x-1) = T(x/2).

Then I think one of the A(x) satisfies

A(x) - T(x) = O(T(x)^C)

where 0<C<1

and also

lim x-> +oo A(x)/T(x) = 1.

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I think I have seen 2d cellular automaton that give the Original sequences or approximated the functional equations given here.

I believe they came from Wolfram and Cook but Im not sure.
Anyone else who might recall this ?

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As for J(x) <=> J ' (x) = J(x/2)
I think I know a way to prove lim x-> +oo A(x)/J(x) = 1.083...

But what really intrests me is a closed form for 1.083...
Havent I seen this number before ??


regards

tommy1729

" Choice is an illusion " tommy1729