Length of a Cartoony Round Cloud
In this post, we again revisit the cartoony round cloud, for which we found the Cartesian equation nine years ago and the area two days ago. Here, we ask ourselves to take some steps...
Maths, Physics and Apps by Stefano Spezia Blog
Area of a Cartoony Round Cloud
In this post, we revisit the cartoony round cloud, for which we found the Cartesian equation nine years ago. Here, we ask ourselves to see what general formula is for the area of such...
On the counting of k-digit terms in increasing sequences of positive integers
In this article, we prove that the sequence of the number of k-digit terms in an increasing sequence of positive integers in a given base , satisfies the following limit of the ratios ,...
Laplace Transforms and the Conversion between Ordinary and Exponential Generating Functions
In this brief article, we prove two useful formulas for converting an ordinary generating function (OGF) of a sequence into the corresponding exponential generating function (EGF), and vice versa. These formulas utilize Laplace integral...
On the positive definiteness of a binary quadratic form
In this brief post, we present a straightforward proof of the positive definiteness of a real binary quadratic form. Theorem. A real binary quadratic form is positive definite if and only if , ,...
Number of correct decimal digits of Pi obtained from the sequences A380099 and A380100
References OEIS Foundation Inc. (2025), Entries A380099 and A380100 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/
A determinantal formula for the number of trees on n labeled nodes
References OEIS Foundation Inc. (2023), Entries A000272, A362679, and A362711 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/
Lecture slides: “Witch of Agnesi and the Gaussian curve”
©Stefano Spezia. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Vertical semicircular guide uniformly accelerated
Problem text A vertical semicircular guide, with radius and without friction, is constrained to a platform that moves with constant acceleration along the horizontal direction (see figure). A point body of mass , initially...
Conic sections: an introduction from space
The conic sections (or simply conics) are those curves that are obtained as the intersection of the surface of a cone with a plane. Here, we discuss what kind of curves are achieved by varying the characteristics of the plane...
