25 problems

Let $T_{i,j}$ denote the entry in row $i$ and column $j$ of the Tchoukaillon array. Then, as $i+j\to\infty$ with $i,j\ge 0$, …

Let $c1_{2,3}(s,n)$ denote the quantity appearing in the preceding conjecture, with $s$ fixed and $n$ varying. The recurrence conjecture. … The source explicitly says that, using t…

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…

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

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…

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

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

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…

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

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,

Consider a family of matrices to which the Laplace expansion process described in the source is applied. Convergence conjecture. For matrix families satisfying specified criteria,…

Let $p=q=1/2$, and let $W_{n,0,0}(x)$ be the probability generating function for the two-player pile game with boundary, with $n$ the initial pile size. Define the first player's w…

Let $a_n(\beta)$ be defined by … with $a_0=1$, $a_1=0$, and $a_2=\beta$, and let $h_n(\beta)$ be its Hankel transform. Recurrence-form Hankel-transform conjecture. The Hankel trans…

Let $T$ be a set of relatively prime pairs, and let $(\alpha,\beta)\in R_T'$, where … Let $g_t^{\alpha,\beta}$ denote the sequence associated with the $(\alpha,\beta)$-walk, and le…

Characterization conjecture. There exist infinitely many $n$ with $(\alpha,\beta)\in\mathbf{S}_T(n)$ if and only if $(\alpha,\beta)\in R_T'$. This concerns which relatively prime p…

Let $G_s(n,k)$ be the recurrence obtained by summing over partitions of $k$ having size at most $s$, with $q$ an indeterminate and $n,k,s$ in the domain where this recurrence is de…

Gessel-sequence interpretation conjecture. The sequence $b_n$ has a combinatorial interpretation of type (C1).

Let $\mathcal{I}_n(n)$ denote the number of the relevant perfect lattice paths in an $n\times n$ table. Povolotsky's recurrence conjecture. For positive integers $n$, the following…

An ordered partition of $[2k]$ with $k$ blocks of size $2$ is counted by $op_{[2^k]}(123)$ when it avoids the permutation pattern $123$. Godbole–Goyt–Herdan–Pudwell's recurrence co…

Let $C_n^{(2)}(q,t)$ denote the generalized $(q,t)$-Catalan number of level $2$. Level-2 recursion conjecture. For $n>1$, … The formula is presented as a conjectural special-case d…