arXiv subjects
Publications and source records attributed to Greg Dresden.
For the Fibonacci numbers $F_n$, we have the self-convolution formula $5 \sum_{i=0}^n F_i F_{n-i} = (2n)F_{n+1} - (n+1)F_n$. We find the corresponding self-convolution formula for the Narayana numbers $R_n$ which satisfy $R_n = R_{n-1} + R_{n-3}$, and then generalize it to the $k$-step Narayana numbers $\mathcal{R}_n$ with order-$k$ recurrence formula $\mathcal{R}_n = \mathcal{R}_{n-1} + \mathcal{R}_{n-k}$.
We take Conway's Look and Say Sequence into a base-3 world, and we discover that there are only 24 interesting and irreducible sequences in base 3.
Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.
We give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sums of coefficients along certain diagonals, we obtain the Fibonaccis, the Tribonaccis, the Tetranaccis, and so on.
Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form $a^{n}+1$ can be expressed as the sum of two integer squares. We prove that $a^n + 1$ is the sum of two squares for all $n \in \mathbb{N}$ if and only if $a$ is a perfect square. We also prove that for $a\equiv 0,1,2\pmod{4},$ if $a^{n} + 1$ is the sum of two squares, then $a^δ + 1$ is the sum of two squares for all $δ| n, \ δ>1$. Using Aurifeuillian factorization, we show that if $a$ is a prime and $a\equiv 1 \pmod{4}$, then there are either zero or infinitely many odd $n$ such that $a^n+1$ is the sum of two squares. When $a\equiv 3\pmod{4},$ we define $m$ to be the least positive integer such that $\frac{a+1}{m}$ is the sum of two squares, and prove that if $a^n+1$ is the sum of two squares for any odd integer $n,$ then $m | n$, and both $a^m+1$ and $\frac{n}{m}$ are sums of two squares.