Is there a particular sequence of amino acids that we know will take on a beta-sheet or an alpha helix or is it essentially random?
Yes, there are certain amino acid sequences that tend to form alpha-helices, and others that prefer to form beta-sheets. There is no perfect correspondence between sequence and structure, but there is a statistical relation where presence of certain amino acids in particular sequences makes one conformation or the other more likely. For example, alanine, glutamate, leucine and methionine tends to be present in alpha helices.
This topic has been studied for 50 years or so, and there are many methods available for predicting structural features from an amino acid sequence. See this list of available software, and this article about protein structure prediction in general.
See this table from another answer on this site which shows which individual amino acid prefers which secondary structure (higher value shows higher preference):