10 problems

Alternating-sign conjecture. Like the Bernoulli numbers, the signs of the even-indexed Bernoulli-Catalan numbers $\tau_{2n}^*$ alternate.

Extension conjecture. The same methods should work for other sequences, including Pell and Tribonacci numbers.

Let the integer sequence be the variant of the happy-numbers sequence defined in the paper, and let $k$ be one of its terms. Three-step convergence conjecture. The integer $k$ conv…

Generalized Fibonacci-ratio convergence conjecture. There is a converging sequence of some generalized Fibonacci ratios, given the difference between consecutive ratios, including…

Let $p^0_i,q^0_i$ and $p^1_i,q^1_i$ enumerate, respectively, all losing positions of the forms $\{0,p^0_i,q^0_i\}$ and $\{1,p^1_i,q^1_i\}$, with each family arranged in increasing…

Let $p^0_i,q^0_i$ and $p^1_i,q^1_i$ denote the coordinates of the losing positions of the forms $\{0,p^0_i,q^0_i\}$ and $\{1,p^1_i,q^1_i\}$, respectively. Here, the pairs are compa…

Fix $(p,q)=(1,2)$ or $(p,q)=(2,3)$, and let $(a_n,b_n)$ be the associated sequences. Two sequences $(l_i)$ and $(u_i)$ are complementary when they are increasing and together parti…

Fix $k\in\mathbb{N}_0$ and let $Q=((p_i,q_i))_{i\in\{0,1,\ldots,k\}}$. Let $\pi_Q$ be the permutation associated with $Q$ as defined earlier, and let a $C$-fold split have the mean…

Fix $p,q\in\mathbb{N}$, let $\pi=\pi_{p,q}$, and let $a=a(p,q)$ and $b=b(p,q)$ be the associated sequences. Let $U$ and $\mathbb{N}_0$ have the meanings fixed earlier in the paper,…

Fix $p,q\bin \mathbb{N}$, and let $a=a(p,q)$ and $b=b(p,q)$ denote the associated sequences. A pair $(p,q)$ is a splitting pair if it is a Wythoff pair or a dual Wythoff pair. The…