07/27/2014, 08:44 PM

My favorite integer sequence is http://oeis.org/A018819

or resp http://oeis.org/A000123

1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60 etc

the idea that they (A018819) are computed as f(1)=f(2)=1 , f(2m-1) = f(1)+f(2)+...f(m) = f(2m) fascinates me.

This is a nicer sequence/equation than Fibonacci and Somos or even Sylvester imho.

Also very appealing is that the second difference of any of these is itself and the first difference is the other one.

This seems like a natural analogue or extension of D exp(x) = exp(x) or the hyperbolic case (sinh,cosh) or even the differences of 2^n are 2^n.

But it does not grow exponential in the sense that A^n is not a good fit for f(n).

Its growth rate is in between polynomial and exponential what considering the differences is a bit counterintuitive (?).

This is the simplest functional equation for strictly nondecreasing integer sequences with a growth rate between polynomial and exponential that I know.

I wonder if this can be extended naturally to a real-analytic function !?

I assume this is already done if possible , but I cant recall immediately.

Perhaps the equation Dif^[2] (f(z)) = f(z+q) helps.

(Dif is the difference operator and q is a real number)

Generalizations of this are awesome too.

No proven connection with tetration from these generalizations has been shown yet so I post it here and not in the General math section.

Continu sums and continu repeated sums are a part of these generalizations.

It seems that the generalizations of this not so well known sequence are also hidden from most books , just like tetration or cellular automatons are not in the complex analysis books or any other standard book.

Still digging for the hidden fundamentals of math

regards

tommy1729

or resp http://oeis.org/A000123

1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60 etc

the idea that they (A018819) are computed as f(1)=f(2)=1 , f(2m-1) = f(1)+f(2)+...f(m) = f(2m) fascinates me.

This is a nicer sequence/equation than Fibonacci and Somos or even Sylvester imho.

Also very appealing is that the second difference of any of these is itself and the first difference is the other one.

This seems like a natural analogue or extension of D exp(x) = exp(x) or the hyperbolic case (sinh,cosh) or even the differences of 2^n are 2^n.

But it does not grow exponential in the sense that A^n is not a good fit for f(n).

Its growth rate is in between polynomial and exponential what considering the differences is a bit counterintuitive (?).

This is the simplest functional equation for strictly nondecreasing integer sequences with a growth rate between polynomial and exponential that I know.

I wonder if this can be extended naturally to a real-analytic function !?

I assume this is already done if possible , but I cant recall immediately.

Perhaps the equation Dif^[2] (f(z)) = f(z+q) helps.

(Dif is the difference operator and q is a real number)

Generalizations of this are awesome too.

No proven connection with tetration from these generalizations has been shown yet so I post it here and not in the General math section.

Continu sums and continu repeated sums are a part of these generalizations.

It seems that the generalizations of this not so well known sequence are also hidden from most books , just like tetration or cellular automatons are not in the complex analysis books or any other standard book.

Still digging for the hidden fundamentals of math

regards

tommy1729