26 problems

Recurrence and coefficient conjecture. The linear recurrence corresponding to $k$ has order $\left\lfloor k/2\right\rfloor+3$, and each coefficient $c_j(q)$ is a linear polynomial…

Direct recurrence conjecture. The coefficients can be computed directly by

Nonexistence conjecture. The equation has no integer solutions $(x,n,m)$ for $x>1$ and $n\geq 1$ when $x\equiv 0,1,$ or $3\pmod 4$. Equivalently, the corresponding Diophantine equa…

Recurrence-product conjecture. The sequence satisfies

Let $p_{n+1}(z)=\sum_{i=1}^{k}Q_i(z)p_{n-i}(z)$ be a finite recurrence with fixed polynomial coefficients, and suppose $\deg p_n(z)=n$ for $n\in\mathbb N$. Let $\Xi_Q$ be the equim…

For ternary partitions, let $\operatorname{num}_{\mathcal T}(n,x)$ denote the associated numerator. Define $t(n)\coloneq\operatorname{num}_{\mathcal T}(n,1)$ for $n\geq 0$, with…

Arm-property conjecture. The identity

Let a family of graphs have Laplacian matrices, and apply the procedures LaplaceExpand and SystemReduce together with the mathematical theorems described in the source. The numerat…

Let $\operatorname{Pol}_n(x)$ be the monic polynomial of degree $2n-2$ occurring in the factorization of the determinant in Theorem MS1. Polynomial-factor recurrence conjecture. Th…

Let $C^X_{r,j}$ denote the entries of the $X$-circuit array. For $c=1,r=0$ and for each $c \ge 2$ with corresponding $r \in \{2(c-1),2(c-1)-1\}$, suppose there is a sequence of pol…

Let $C^X_r(j)$ and $C^M_r(j)$ denote the entries in rows $r$ of the corresponding circuit arrays, and let $Num$ and $Den$ denote their numerators and denominators. A sequence satis…

Let $(d_n)$ be the unique sequence of rational numbers with initial terms … satisfying the explicit order-seven linear recurrence … where each $r_i(n)\in\mathbb{Z}[n]$ is defined b…

Asymptotic growth conjecture. Any solution to this system grows asymptotically like

Let $X(\eta)=I\Xi_{\mathcal{D}}Y$ have degree $L$, and let $r_{n,k}^{X,\mathcal{D}}$ be the coefficients in the $1+2L$-term constant-coefficient recurrence for the multi-indexed Me…

Let $P_{\mathcal{D},n}(\eta;\boldsymbol{\lambda})$ be the multi-indexed Meixner–Pollaczek or continuous Hahn polynomial, with parameters $\boldsymbol{\lambda}$ and index set…

Sahin's extended conjecture. For any positive integers $r\leq d$, we have

For integers $s$, $r$, and $d$ with $1\le r\le d$, let $N_{d,r}(s)$ denote the number of $(s,s+r)$-core partitions with $d$-distinct parts. Recurrence conjecture. The values of…

At-most-two non-real zeros conjecture. Every polynomial $W_n(z)$ has at most two non-real zeros.

Charleston and Steel's conjecture. For $T_k$ and $r\geq 2$,

Conjectured recurrence relations. The numerical data in the table suggest recurrence relations for the maximal terms of the TRIP-Stern sequences corresponding to the remaining 96 m…

Finite-versus-infinite conjecture. The number of order $p$ $(\alpha,\beta)$-Conolly meta-Fibonacci recursions is finite if $\beta>0$ and infinite if $\beta=0$.

Classification conjecture. For $\beta>0$, the only 2-ary order $2$ $(\alpha,\beta)$-Conolly recurrences are exactly the recurrences listed in the source for $(\alpha,\beta)=(-2,3)$…

Positivity and uniqueness conjecture. There is a unique positive solution, and it corresponds to the initial condition

Gazeau et al.'s conjecture. These numbers satisfy the recurrence relation

Closure conjecture. Multivariate $q$-holonomic sequences are closed under twisting by complex roots of unity and under substitutions of the form $q\to q^{\alpha}$ for…