My favorite integer sequence

[tommy1729]

In post 19 I wrote :

quote:
5) I would like to comment that f(x) - f(x-1) can Always be approximated by
f(x) - ( f(x) - f ' (x) + f '' (x)/2 - f "' (x)/6 + f ""(x)/24 )
(end quote)

Jay tried to approximate the Original sequence(s) by the equation

f ' (x) = f(x/2)

A better approximation is probably

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

It seems to hard at first , however

f(x) - f(x-1) can Always be approximated by
f(x) - ( f(x) - f ' (x) + f '' (x)/2 - f "' (x)/6 + f ""(x)/24 )

Therefore we can rewrite

f ' (x) - f " (x)/2 + ... = f(x/2).

These coefficients can be solved for by using the binomium expansion of the powers in the Taylor series for f(x-1).

This gives us a linear system of equations.

Actually we do not need the equation with the derivatives , its just equivalent. Just expand (x-1)^n and solve for the Taylor coefficients.

SO

(a0 + a1 x + a2 x^2 + ...) - (a0 + a1 (x-1) + a2 (x-1)^2 + ... ) =
a0 + a1/2 x + a2/4 x^2 + ...

Get rid of the (x-1)^n terms by using the binomial theorem.
Solve the system of equations.

regards

tommy1729