Search Results

Here is partial answer which might give infinitely many cheap solutions in some base. If I am not mistaken, there are no cheap solutions in bases up to $100$. Let $f_n(x)$ be the sum of $n$ consecuti …

Let $a(n)$ be linear recurrence with constant coefficient of order $t$. Assume $a(n)=\sum_{i=0}^t c_i r_i^n$ where $r_i$ are the roots of the companion polynomial and $c_i$ are algebraic numbers. Th …

Let $a_n$ be a linear recurrence with integer constant coefficients and initial values. Is it possible $a_n$ to satisfy all of these: $a_n = 0$ infinitely often. if $a_n \ne 0$, $ | a_n |$ is of ex …

$F_n$ are the Fibonacci numbers. In On computing factors of cyclotomic polynomials p.1 for odd square-free $n>1$ the cyclotomic polynomial $\Phi_n(x)$ satisfies: $$ 4 \Phi_n(x)=A_n(x)^2 - (-1)^{(n-1)/ …

By Robin's theorem $$G(n)=\frac{\sigma(n)}{n \log \log n}$$ is bounded by $e^\gamma \approx 1.78107241799$ for $n>5040$ assuming Riemann hypothesis . For $n=\mathrm {lcm} (1,2 \dots k)$, $G(n)$ app …

Numerical evidence suggests the following. For $c \in \mathbb{N}, c > 2$ define the sequence $a_n$ by $a_0=0,a_1=1, \; a_n=c a_{n-1} - a_{n-2}$ For $ 5 < n < 500, \; 2 < c < 100$ there are no primes i …

Probably this is well known. $\theta_2$ and $\theta_3$ are Jacobi theta functions as defined in mathworld (31) and (32). For natural $n$ define $$ f(n) = \frac{\theta_2(-e^{-\pi\sqrt{n}})}{\theta_3(-e …

This might be related to an open problem. For odd natural $n$ define the Euler quotient: $$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$ Q1 Are there infinitely many $n$ …

For real irrational $C > 1 $ and natural $n,b$, define $a(C,n,b)=\lfloor C^n \rfloor \mod b$ Q1 For which $C,b$ is $a(C,n,b)$ computable in time polynomial in $\log{n}$? Searching in OEIS sug …

Related to this question. Let $p$ be prime and $n$ positive integer. Define $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$ Let $D(n)$ be the base $2$ discrete logarithm of $a(n)$, i.e. given $p,a(n)$ we have $2 …

This is related to problem in graph theory. OEIS defines A033485 as $a(1)=1$ and $a(n)=a(n-1)+a(\lfloor n/2 \rfloor)$. Q1 what are upper bounds and asymptotics for $a(n)$, can we get $\exp(o(n))$? F …

For $b_k$: Computing the first few terms and searching into https://oeis.org returns A218222 and A088716. A088716 is very close to your sequence. If $b_k$ is really A218222, from the comments: $$ …

Edit Aaron solved the original question with the fourth order $$ a(n)=n2^n+\frac{(-1)^n-1^n}{2} $$ trying to make the question harder. Let $a(n)$ be a linear recurrence with constant coefficients, of …

Search in OEIS Returns 3 results. Most likely apart from the initial term it is A042950 G.f.: (2-x)/(1-2*x) a(n)=2*a(n-1), n>1; a(0)=2, a(1)=3.

Background: The binomial coefficients $C(n,k)$ satisfy the recurrence $C(n,k)=C(n-1,k)+C(n-1,k-1)$ and some terminating conditions, for more information check here. $C(n,k)$ doesn't appear to be effi …