6 problems

Let $k>3$, and let $F^{(k)}=(F_n^{(k)})_{n\geq 2-k}$ be the $k$-Fibonacci sequence defined by … for $n\geq 2$, with $F_{-(k-2)}^{(k)}=\cdots=F_0^{(k)}=0$ and $F_1^{(k)}=1$. A repdi…

Fix $k\geq 2$ and define $H_{m,a}^{(k)}(x)$ as the generating function counting coefficients of $\prod_{i=1}^n(1+x^{F_{i+k-1}^{(k)}})$ that are congruent to $a$ modulo $m$. Parity-…

Fix $k\geq 2$, $m\geq 1$, and $0\leq a<m$. Set … Let $h_{m,a}^{(k)}(n)$ be the number of coefficients of $I_n^{(k)}(x)$ congruent to $a$ modulo $m$, and define … Congruence-count r…

Fix $r\geq 2$ and $k\geq 2$, and let $v_r^{(k)}(n,t)$ be the sum of the $r$th powers of the coefficients of … with … Denominator-shape conjecture. There is an integer $m\geq 1$, de…

Fix $k\geq 2$ and let $v_3^{(k)}(n,t)$ be the sum of the cubes of the coefficients of … Define … Third-moment generating-function conjecture. One has … where … This conjecture give…

Generalized Fibonacci analogue conjecture. The generating function