7
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Fact 1: Given the eta quotient,

$$x_d= e^{2\pi\rm{i}/48}\,\frac{\eta(\tau)}{\eta(2\tau)}$$

where $\tau=\frac{1+\sqrt{-d}}2$, then $x_d$ for $d=11,19,43,67,163$ are the roots of the simple cubics,

$$x^3-2x^2+2x-2=0\\x^3 - 2x - 2 =0\\ x^3 - 2x^2 - 2=0\\ x^3 - 2x^2 - 2x - 2=0\\ \color{blue}{x^3 - 6x^2 + 4x - 2=0}$$

Fact 2: The number of Hamiltonian cycles in $C_5 \times P_n$ (OEIS A003731) is,

$$a_n = 1, 5, 30, 160, 850, 4520, 24040, 127860, 680040, 3616880, 19236840\dots$$

and, for $n>3$, has recurrence

$$\color{blue}{a_n -6a_{n-1} + 4a_{n-2} - 2a_{n-3}=0}$$

Of course, the ratio of consecutive terms tend to $x_{163}$. For example,

$$\begin{aligned}\frac{19236840}{3616880}&=5.318628210\dots\\ x_{163}&=5.318628217\dots\end{aligned}$$

Q: Is the appearance of the blue polynomial in graph theory a coincidence, or does it have to do with $Q\sqrt{-163}$ being a UFD? Do the other polynomials have a role for other Hamiltonian cycles $C_m\times P_n$?

P.S. Fortunately the OEIS has an index of linear recurrences. A check showed that all of the polynomials above have one or more associated sequences (some of them seem to be tantalizingly related).

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    $\begingroup$ This is really nice! But can you specify what $P_n$ is? $\endgroup$ – Vincent Dec 13 '17 at 15:29
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    $\begingroup$ @Vincent: I think it's a path graph. The sequence is by Frans Fasse and his website might be helpful. $\endgroup$ – Tito Piezas III Dec 13 '17 at 15:38
  • $\begingroup$ I went through Fasse's site more thoroughly and his results for $C_4 \times P_n$ involve polynomials that are not analogous. $\endgroup$ – Tito Piezas III Dec 13 '17 at 16:29
  • $\begingroup$ @TitoPiezasIII Although that particular $\eta$-quotient is not a McKay-Thompson series for Monstrous Moonshine, have you looked at similar results for them? Or $\eta$-quotients in general? $\endgroup$ – Somos Dec 13 '17 at 19:23
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    $\begingroup$ I'm not a graph theorist, but I feel like you get loads of recurrences by looking at various counting problems in graph theory, certainly way more than the number of imaginary quadratic rings with class number 1 (there are 13). It's possible there's something going on, but I don't feel that just comparing a few recurrences to a few polynomials is strong evidence for anything. $\endgroup$ – Kimball Dec 14 '17 at 3:26

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