template <typename T> auto square(T x) { return x * x; }

template <typename T> auto gaussian(size_t N, T t_width) {
	return [=](auto x) { return exp(-square((x - (((T)N) - 1) / 2) / (2 * t_width))); };
}

template <typename T> auto approximate_confined_gaussian(size_t N, T t_width) {
	auto G = gaussian(N, t_width);
	return [=](auto n) {
		return G(n) -
		       (G(((T)-0.5)) * (G(n + N) + G(n - N))) / (G(((T)-0.5) + N) + G(((T)-0.5) - N));
	};
}