08/05/2014, 11:16 PM

Some Top posters of the tetration forum :

Bo " administrator " 198214

Tommy " sinh " 1729

Gottfried " matrix " Helms

Jay "excel" fox

sheldon " theta wave "

mike "continuum sum " 3

Jms " transform " Nxn

Ive been wanting to post that silly thing for a long time

Anyway the posts made by Gottfried and Jay look familiar to me.

I also used excel to accelerate the computation of the sequence and investigate its number theoretic properties.

I Always factor number sequences.

Anyways a few remarks/ideas :

1) Many years ago I mentioned the equations

A) f ' (2x) = 3 f(x) + C

B) f ' (x) = f(x/2) + f(x/3) + C

at , if im not mistaken , sci.math.

Seeing Jay's f ' (2x) = f(x) makes me wonder about those again.

Unfortunately I have holes in my brain.

2) Our sequence satisfies f(n) - f(n-1) - (f(n-1) - f(n-2)) = f(n-2) or similar if we IGNORE ZERO. I havent quite made that clear before.

3) What I was really looking for is a closed form expression for the sequence.

Some transformation or such like for the gamma function.

4) I would like a single functional equation for the series that is exact and does NOT use rounding. If possible.

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 )

Hence differences / difference equations can be approximated by differential / differential equations.

This might be usefull as it has been in the past.

Maybe it can connect Jay's f(x) with the Original sequence , maybe express one in terms of the other or solve 3) or 4).

Those are the most important things for me.

regards

tommy1729

Bo " administrator " 198214

Tommy " sinh " 1729

Gottfried " matrix " Helms

Jay "excel" fox

sheldon " theta wave "

mike "continuum sum " 3

Jms " transform " Nxn

Ive been wanting to post that silly thing for a long time

Anyway the posts made by Gottfried and Jay look familiar to me.

I also used excel to accelerate the computation of the sequence and investigate its number theoretic properties.

I Always factor number sequences.

Anyways a few remarks/ideas :

1) Many years ago I mentioned the equations

A) f ' (2x) = 3 f(x) + C

B) f ' (x) = f(x/2) + f(x/3) + C

at , if im not mistaken , sci.math.

Seeing Jay's f ' (2x) = f(x) makes me wonder about those again.

Unfortunately I have holes in my brain.

2) Our sequence satisfies f(n) - f(n-1) - (f(n-1) - f(n-2)) = f(n-2) or similar if we IGNORE ZERO. I havent quite made that clear before.

3) What I was really looking for is a closed form expression for the sequence.

Some transformation or such like for the gamma function.

4) I would like a single functional equation for the series that is exact and does NOT use rounding. If possible.

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 )

Hence differences / difference equations can be approximated by differential / differential equations.

This might be usefull as it has been in the past.

Maybe it can connect Jay's f(x) with the Original sequence , maybe express one in terms of the other or solve 3) or 4).

Those are the most important things for me.

regards

tommy1729