91Ó°ÊÓ

Let \(X\) be the set of positive integers that are not perfect squares. (A perfect square \(m\) is an integer of the form \(m=i^{2}\) where \(i\) is an integer.) Concern the sequence s from \(X\) to \(\mathbf{Z}\) defined as follows. If \(n \in X,\) let \(s_{n}\) be the least integer \(a_{k}\) for which there exist integers \(a_{1}, \ldots, a_{k}\) with \(n

Let \(X\) be the set of positive integers that are not perfect squares. (A perfect square \(m\) is an integer of the form \(m=i^{2}\) where \(i\) is an integer.) Concern the sequence s from \(X\) to \(\mathbf{Z}\) defined as follows. If \(n \in X,\) let \(s_{n}\) be the least integer \(a_{k}\) for which there exist integers \(a_{1}, \ldots, a_{k}\) with \(n

Prove that if \(n\) is an odd integer, $$ \left.\mid \frac{n^{2}}{4}\right\rfloor=\left(\frac{n-1}{2}\right)\left(\frac{n+1}{2}\right) $$

Prove that if \(n\) is an odd integer, $$ \left[\frac{n^{2}}{4}\right]=\frac{n^{2}+3}{4} $$

List all strings over \(X=\\{0,1\\}\) of length 3 or less.

Find all substrings of the string \(b a b c\)

Find all substrings of the string aabaabb.

Let \(L\) be the set of all strings, including the null string, that can be constructed by repeated application of the following rules: If \(\alpha \in L,\) then \(a \alpha b \in L\) and \(b \alpha a \in L .\) If \(\alpha \in L\) and \(\beta \in L,\) then \(\alpha \beta \in L\) For example, \(a b\) is in \(L\), for if we take \(\alpha=\lambda,\) then \(\alpha \in L\) and the first rule states that \(a b=a \alpha b \in L\). Similarly, \(b a \in L\). As another example, aabb is in \(L\), for if we take \(\alpha=a b\), then \(\alpha \in L\); by the first rule, \(a a b b=a \alpha b \in L .\) As a final example, aabbba is in \(L\), for if we take \(\alpha=a a b b\) and \(\beta=b a,\) then \(\alpha \in L\) and \(\beta \in L ;\) by the second rule, \(a a b b b a=\alpha \beta \in L\). Show that \(a a b\) is not in \(L\).