20 problems

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

Recurrence conjecture. The sequence $(\tau_{n,r})$ satisfies the linear recurrence of order $2r+2$

Let $\{s_n\}_{n \in \mathbb{N}}$ be a sequence satisfying a particular convolution recursion, and let $\{H_n\}_{n \in \mathbb{N}}$ be its sequence of Hankel determinants. Barry's c…

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…

Somos 4 conjecture. The Hankel transform of the expansion of $f(x)/x$ is a

Exact valuation conjecture. For the relevant integers $m$ and $l$, one has

A Somos-$k$ sequence is a sequence defined by a higher-order Somos recurrence. A discrete linear flow on the Jacobian of a hyperelliptic curve is a discrete dynamical evolution giv…

Let a Somos sequence be a sequence of the type considered in the source, and let an appropriate divisor sequence on a curve be the associated sequence obtained from divisors on suc…

Let $(A_n)$ be the Somos-4 sequence referred to in the source, and for a prime $p$ let the gap length be the distance between successive terms of the sequence divisible by $p$. Rob…

Let $\tau(n_1,\ldots,n_s)=\langle g\mid n_1,\ldots,n_s\rangle$ be the discrete function defined for integer tuples with $\operatorname{deg}(\mathbf n)=\sum_i n_i=0$. For…

Hankel determinant periodicity and recurrence conjecture. The $k+2$ sequences $\Delta_n^{(0)},\Delta_n^{(1)},\dots,\Delta_n^{(k+1)}$ take only the values $-1,0,1$. They are…

Let $E=E(a,b,c,d)$ be the elliptic curve … with nonzero discriminant, and let $(0,0)$ be the reference point. Let $h_n$ be the Hankel transform of the sequence generated by … where…

Consider the original Somos-4 sequence with $x_{-1}=x_0=x_1=x_2=1$, and let $y_n^{(iv)}$ be the fourth independent shadow sequence obtained by solving the associated homogeneous eq…

General Somos 4 conjecture. The Hankel transform is an $(\alpha,\beta)$ Somos $4$ sequence, where

Let $a_n$ be the sequence with generating function … where $c(x)$ is the Catalan generating function. Specialized elliptic-curve Somos 4 conjecture. The Hankel transform of $a_n$ i…

Let $a_n(p,q,r,s,t)$ be the five-parameter generalized Catalan sequence. Assume … Specialized generalized Catalan Somos 4 conjecture. The Hankel transform of $a_n(p,q,1,q-p+1,1)$ i…

Let $a_n(p,q,r,s,t)$ be the five-parameter generalized Catalan sequence and let $h_n$ denote the Hankel transform of $\frac{1}{p}a_{n+1}(p,q,r,s,t)$. A $(\alpha,\beta)$ Somos $4$ s…

Integrality conjecture. The sequence $(b_n)_{n\in\mathbb{N}}$ is integer if and only if $p=1$; analogously, for the second recurrence in the preceding corollary, integrality holds…

Let $b_0=1$, $b_1=\alpha'$, and, for $n\geq 2$, define … where $\alpha'$, $\beta'$, and $\gamma'$ are constants. Let $H_n^{(m)}$ be the Hankel determinant … with $H_0^{(m)}=1$, and…

Let $T$ be the recurrence tree generated by the recurrence referred to as (Som4Ord3), and let the generalized Somos-4 sequence be the sequence associated with that recurrence. Prod…