6 problems

Equality conjecture.

Let $b_0=0$, $b_1=1$, and, with $\tau=\sqrt{2}$, define … for $n\geq 0$. Here $\log_3$ denotes the logarithm to base $3$. Northshield's conjecture. … Northshield proved the corresp…

Let $(s(n))_{n\ge 0}$ be Stern's sequence, defined by $s(0)=0$, $s(1)=1$, $s(2n)=s(n)$ and $s(2n+1)=s(n)+s(n+1)$. Let $L_m(r)$ denote the $m$th largest distinct value in the $r$th…

Let $e(n)$ denote the Stern sequence, and for a nonnegative integer $k$ define … The function $\mathcal{E}_{k}(x)$ is transcendental for $k\geq 2$. This extends the known facts tha…

Generating-series factorization conjecture. The series $A(z)$ satisfies

Let $s(n)$ be the Stern sequence, let $t(0)=0$, $t(1)=1$, and define $t(2n)=-t(n)$ and $t(2n+1)=-t(n)-t(n+1)$ for $n\geq 1$. For a formal variable $z$, write the corresponding gene…