Define $g(x)$ as :
If $f(m) =< x < f(m+1)$ for a positive integer $m$ then $g(x) = m$.
Now we define $f(n)$ for strict positive integer $n$.
$$f(1) = 1 $$ $$f(2) = 3 $$ $$f(3) = 7 $$
For $n > 3 $ we have
$$ f(n) = f(n - 1) + f( n - g(n) ) $$
Thus
$$ f(4) = f(3) + f(4 - g(4)) = f(3) + f( 4 - 2) = f(3) + f(2) = 7 + 3 = 10. $$ $$f(5) = f(4) + f(5 - 2) = f(4) + f(3) = 10 + 7 = 17 $$
The sequence we get from $f(n)$ is
$1,3,7,10,17,27,37,54,81,108,145,199,280,388,533,732,931,1211,...$
It is easy to show that this sequence eventually grows slower than any exponential.
Hint : compare the functional equation with this $t(n) = t(n-1) + t(n-k) $ ( see notes * )
It is also easy to show that this sequence eventually grows faster than any polynomial.
In fact faster than $exp(ln(n)^r) $. Hint : Compare the functional equation with $d(n) = d(n-1) + d(floor(n/k)) $
A good estimate for $f(n)$ might be $C_1 \space \exp(n^A + \exp(\ln(n)^B)) \space \exp(\ln(n)^C) \space n^D $ for appropriate values $C_1,A,B,C,D$ ( notice $B < 1 $)
Or maybe $\exp(n^A) \ln(n)^B $ or ... How to even guess the best form ?
It might help to add special functions like Lambert-W , for instance investigate asymptotics with lambert-W or consider $h’(x) = h( x - W(x)^y ) $ for Some $ y > 1 $ as a potential asymptotic for $f(n)$.
This sequence $f(n)$ fascinates me. And I wonder about its upper bounds , lower bounds and asymptotics.
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Notes
I assume the sequence contains infinitely many primes , but proving that is probably beyond human powers , so i will not even ask for it.
- This is related to fibonacci and tribonacci. We might want to relate the equation discussed here ( to estimate the growth ) as it might help to find an asymptotic for $f(n) $ ?? See :
Asymptotic to a sequence of algebraic numbers.
Also a similar recursion with slower growth has been discussed here and on OEIS see :
Asymptotics for the sequence A094589
Perhaps that might help.
Nice plots are Nice :)
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