Michael Lugo
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 Jan 19 answered What's the period of $\sin \left(x^\frac{3}{2}\right)$? Jan 19 comment Will you eventually run out of fair coins which either triplicate or disappear when flipped? You're describing something an awful lot like a Galton-Watson process: en.wikipedia.org/wiki/Galton–Watson_process Jan 18 answered Why is $\cos^2(2\pi/5) + \cos^2(4\pi/5)=3/4$? Jan 13 revised Why does this sequence converge to $\pi$? fixed formatting in second line Jan 13 comment Why does this sequence converge to $\pi$? It appears that $s_k \approx \cos ((k/n) (\pi/2))$. I'd guess that the recurrence here is an approximate numerical solution to some differential equation. Jan 13 comment What is the average number of connected components in a Secret Santa graph? Something seems off here - the average number of cycles in a permutation of $[n]$ is $H_n$. I think there's an extra factor of $1/e$ in your answer and the average number of cycles in a derangement of $[n]$ is close to $H_n-1$. Jan 13 comment Choose a composition from the previous composition Hint: can you use multiple kinds of bars? Jan 12 comment Evaluation of $\displaystyle \int_{1}^{3}\left[\sqrt{1+(x-1)^3}+(x^2-1)^{\frac{1}{3}}\right]dx$ I agree with Lucian. In a sense this is a question about the geometry of curves in the plane - the integral, which turns out to be 6, can actually be written as the area of a single rectangle. Jan 12 answered Make $f(x)=\sin x-\frac{x+ax^3}{1+bx^2}$ be the infinitesimal of the highest order Jan 11 comment Least value of $C$ for which $CT_n+1$ is always a triangular number. Hint: an integer $m$ is triangular if and only if $8m+1$ is square. Jan 11 comment Probability that a quadratic equation with random coefficients has real roots We can generate a random point $(u, v, w)$ on the sphere by taking $X, Y, Z$ independent random standard normals and then taking $(u, v, w) = (X, Y, Z)/\sqrt{X^2+Y^2+Z^2}$. I simulated $10^6$ points on the sphere and found that $351503$ of them had $u^2 + 2v^2 \le 2w^2$, so the probability is in the neighborhood of $0.352$. This hopefully helps confirm an analytic solution if someone attempts it. Jan 8 answered Number of connected sets of size $k$ containing a given vertex on $\mathbb{Z}^d$ Dec 18 answered Calculate the determinant of the matrices $a_{ij}=\frac{1}{i+j-1}$ and $b_{ij}=\frac{1}{i+j}$? Dec 14 comment How can I create 14 unique team rosters from 24 students in groups of 2? That's an interesting question, designing scheduling algorithms that are robust to students dropping. I wish I knew something about this! Dec 12 revised How can I create 14 unique team rosters from 24 students in groups of 2? added combinatorial-designs tag Dec 12 answered How can I create 14 unique team rosters from 24 students in groups of 2? Dec 10 answered asymptotic approximation for number of partitions of integer that do contain 1 nor 2 Nov 25 awarded Good Answer Nov 23 comment Number theory primes and congruences: $p^{32} -1$ is divisible by $16320$ A relevant link to the OEIS: oeis.org/A079612. In fact $p^{32}-1$ is divisible by 16320 whenever $p$ has no factors in common with $16320$. Nov 20 comment Verify the polynomial of degree $≤ 4$ is exact: $\int_0^1f(x)dx ≈ {1\over90}[7f(0) + 32f({1\over 4}) + 12f({1\over 2}) + 32f({3\over 4}) + 7f(1)]$. As a warmup, you might try to prove that $\int_0^1 f(x) \: dx = {1 \over 6} [f(0) + 4f(1/2) + f(1)]$ when $f$ is a polynomial of degree 3 or less. (This is Simpson's rule for integration.)