13 problems

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

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 $\tau(n_1,\ldots,n_s)=\langle g\mid n_1,\ldots,n_s\rangle$ 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…

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

Let $t$ be a parameter and let $a_n$ be the sequence whose generating function is … Transformed elliptic-curve Somos 4 conjecture. The Hankel transform of $a_n$ is a $(1,t^2+3t+1)$…

Let $b_n$ be the sequence whose generating function is … arising from the elliptic-curve family $E_t:y^2+4xy+y=x^3+(t-1)x^2+tx$. A $(\alpha,\beta)$ Somos $4$ sequence satisfies … E…

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…

Let $a_n(p,s,t)$ be the generalized Schröder numbers and let $h_n(p,s,t)$ be their Hankel transform. A $(\alpha,\beta)$ Somos $4$ sequence satisfies … Generalized Schröder Somos 4…

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

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…