8 problems

Let $\beta(N)=\beta^+(N)\cdot\beta^-(N)$ be the base-phi expansion of the number $N$. Let $w$ be a word of length $m$, and let $R_{\cdot w}$ be the sequence of occurrences of numbe…

Let $v_m(A,-1)$ denote the second-order recurrent sequence used in the paper, and let $\delta_{j,k}$ denote the Kronecker delta. Let $n$ be a positive odd integer and let…

Odd-indexed Fibonacci and Lucas determinant conjecture. For every positive odd integer $n$, these determinants satisfy

Let $E_{\{n,k\}}$ denote the Lucas-Eulerian numbers, defined by the Lucas analogue of the Eulerian recurrence. For a polynomial $A_{n,k}(x)$, write $\big\{A_{n,k}\big\}$ for its Lu…

Let $\hat{Q}_3(m,-m,n,\ell)$ denote the twisted hybrid weaving knot, where $m$ is a natural number and $\ell\in\{-1,0,1\}$. Let $\operatorname{Det}(L)=|\Delta^L(-1)|$ denote the de…

Let $m$ be a natural number, and let $n$ and $ell$ be parameters of the twisted hybrid weaving knot, with $Omega((m,m,n,\ell);q)$ the symmetric Laurent-polynomial trace term define…

Let $(F_n)$ and $(L_n)$ be the Fibonacci and Lucas sequences, defined by … and … for $n\geq 2$. Let $n,m,r$ be nonnegative integers, with $(n,m)\neq(1,0),(2,0)$, and let $x\geq 4$.…

Melham's conjecture. For any positive integers $n,m$, there is a polynomial $P_{2m-1}(x)$ of degree $2m-1$ with integer coefficients such that