Considering a right triangle with legs \(c_1\) and \(c_2\), hypotenuse \(i\), and altitude \(h\) relative to the hypotenuse, we find the value of the ratio \(x\) between \(c_1\) and \(c_2\) under the condition that...
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...
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...
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 ,...
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...
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 , ,...
References OEIS Foundation Inc. (2025), Entries A380099 and A380100 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/
References OEIS Foundation Inc. (2023), Entries A000272, A362679, and A362711 in The On-Line Encyclopedia of Integer Sequences, http://oeis.org/
©Stefano Spezia. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
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...
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