51 problems

Somos's conjecture. The sequence $\{\Delta_n\}_{n\in\mathbb{N}}$ satisfies half of the Somos-4 recurrence

Kontsevich's conjecture. Every component of every such iterate is a noncommutative Laurent polynomial in $x$ and $y$.

Let $a_1,a_2,\dots$ be a Somos-5 sequence, meaning that every six consecutive terms satisfy … The Somos-5 Laurent positivity conjecture. Every term $a_n$ is a Laurent polynomial wi…

Let $E(n)$ denote the fluctuation term in the decomposition of $Q(n)$ into its main term and fluctuation. Sublinear fluctuations conjecture. The fluctuation term satisfies … This i…

Let $F(m)$ be the frequency sequence and let … For each block, let $m_k$ be an index at which $F(m)$ attains its maximum. Peak location law. The peak locations satisfy … Numerical…

Zagier's conjecture. Zagier's finite search yielded all the integral solutions to this recurrence.

Recurrence conjecture for $a_n$ and $b_n$. The first diagonal entry satisfies

Let $a_{1234}^{(0)}(n)=P(n,1,1)$ denote the number of permutations in $S_n$ avoiding the pattern $1234$. Recurrence conjecture for 1234 avoidance. With initial values…

Let $S$ be a section, let $f$ be a state, and let $(x_0,y_0,t_0)$ be a point in $\overset{\circ}{Y}\cap\mathcal{L}$ in the future of $S$. Assume that the cone $\mathcal{C}$ meets…

Let $1\leq p\leq q\leq r$ and set $k=p+q+r$. Define a sequence by … with initial conditions $a_i=1$ for $0\leq i\leq k-1$. Robinson's integrality conjecture. The resulting sequence…

The sequences Somos-$4$ through Somos-$7$ are defined by the quadratic recurrences … with $n\geq4$ and $a_i=1$ for $0\leq i\leq3$; … with $n\geq5$ and $a_i=1$ for $0\leq i\leq4$; t…

Refined recurrence conjecture. For $n\geq 3$ and $1\leq i\leq n-2$, we have

Let $a(n)$ be the sequence defined by … R.J.Mathar's conjecture is that … The recurrence was recorded as a conjecture in the OEIS in 2014. It is in fact proved in the paper by subt…

Let $Q(n)$ be the sequence and define the clock sequences … Let $\phi_1$ and $\phi_2$ be bounded or slowly varying functions. Dyadic clock structure conjecture. There appear to exi…

Let $E(n)$ denote the fluctuation term and let $\epsilon(n)$ denote a bounded perturbation. Dyadic renormalization conjecture. The fluctuation term approximately satisfies … This h…

Let $Q(n)$ be the sequence, let $\log_2 n$ denote the base-two logarithm, and let $\rho(n)$ be a smaller correction term. Log-periodic modulation conjecture. There appears to exist…

Let $Q(n)$ denote the sequence defined by the perturbed Hofstadter recursion. Linear growth conjecture. The sequence satisfies … Numerical experiments strongly indicate slope $1/2$…

Dyadic frequency law. The frequencies $F(m)$ satisfy

Recurrence conjecture. The sequence $a_n$ satisfies

Van de Lune's conjecture. The records of $S_n(\sqrt 2)$ satisfy a Pell-like recurrence. The source states that this specific conjecture was confirmed in prior work, so it is includ…

For an irrational number $\xi$, define the deterministic random walk by … Call $r$ a record if $S_r(\xi)$ is different from every preceding partial sum, including $S_0(\xi)=0$. Van…

Let $f_{m,n}$ be the array considered in the example with parameters $\alpha=\beta=\gamma=1$ and initial conditions $f_{n,0}=f_{0,n}=n!$. Write $a_n\approx b_n$ when $a_n-b_n\to 0$…

Let $b_4(n)$ denote the number of balanced $8\times 2n$ matrices. A linear recurrence of order $9$ with polynomial coefficients of degree $36$ should be satisfied by the sequence…

The determinant factorization conjecture. Writing $n=2r$ or $n=2r-1$, respectively,

For a positive integer $n$, let $d_n(0,t)$ satisfy a recursion … where each $c_i(t)$ is a polynomial in $t$, and let $t_0$ be the maximum integer root, if one exists, of…