%0 Journal Article
%T Squares from &lt;i&gt;D&lt;/i&gt;(每4) and &lt;i&gt;D&lt;/i&gt;(20) Triples
%A Zvonko ˋerin
%J Advances in Pure Mathematics
%P 286-294
%@ 2160-0384
%D 2011
%I Scientific Research Publishing
%R 10.4236/apm.2011.15052
%X We study the eight infinite sequences of triples of natural numbers &lt;i&gt;A&lt;/i&gt;=(&lt;i&gt;F&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,4&lt;i&gt;F&lt;sub&gt;2n+3&lt;/sub&gt;&lt;/i&gt;,&lt;i&gt;F&lt;sub&gt;2n+7&lt;/sub&gt;&lt;/i&gt;), &lt;i&gt;B&lt;/i&gt;=(&lt;i&gt;F&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,4&lt;i&gt;F&lt;sub&gt;2n+5&lt;/sub&gt;&lt;/i&gt;,&lt;i&gt;F&lt;sub&gt;2n+7&lt;/sub&gt;&lt;/i&gt;), &lt;i&gt;C&lt;/i&gt;=(&lt;i&gt;F&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,5&lt;i&gt;F&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,&lt;i&gt;F&lt;sub&gt;2n+3&lt;/sub&gt;&lt;/i&gt;), &lt;i&gt;D&lt;/i&gt;=(&lt;i&gt;F&lt;sub&gt;2n+3&lt;/sub&gt;&lt;/i&gt;,4&lt;i&gt;F&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,&lt;i&gt;F&lt;sub&gt;2n+3&lt;/sub&gt;&lt;/i&gt;)  and  A=(&lt;i&gt;L&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,4&lt;i&gt;L&lt;sub&gt;2n+3&lt;/sub&gt;&lt;/i&gt;,&lt;i&gt;L&lt;sub&gt;2n+7&lt;/sub&gt;&lt;/i&gt;), B=(&lt;i&gt;L&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,4&lt;i&gt;L&lt;sub&gt;2n+5&lt;/sub&gt;&lt;/i&gt;,&lt;i&gt;L&lt;sub&gt;2n+7&lt;/sub&gt;&lt;/i&gt;), C=(&lt;i&gt;L&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,5&lt;i&gt;L&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,&lt;i&gt;L&lt;sub&gt;2n+3&lt;/sub&gt;&lt;/i&gt;), D=(&lt;i&gt;L&lt;sub&gt;2n+3&lt;/sub&gt;&lt;/i&gt;,4&lt;i&gt;L&lt;sub&gt;2n+1&lt;/sub&gt;&lt;/i&gt;,&lt;i&gt;L&lt;sub&gt;2n+3&lt;/sub&gt;&lt;/i&gt;. The sequences &lt;i&gt;A&lt;/i&gt;,&lt;i&gt;B&lt;/i&gt;,&lt;i&gt;C&lt;/i&gt;  and &lt;i&gt;D&lt;/i&gt;  are built from the Fibonacci numbers  &lt;i&gt;F&lt;sub&gt;n&lt;/sub&gt;&lt;/i&gt; while the sequences A, B, C and  D  from the Lucas numbers  &lt;i&gt;L&lt;sub&gt;n&lt;/sub&gt;&lt;/i&gt;. Each triple in the sequences &lt;i&gt;A&lt;/i&gt;,&lt;i&gt;B&lt;/i&gt;,&lt;i&gt;C&lt;/i&gt;  and &lt;i&gt;D&lt;/i&gt; has the property &lt;i&gt;D&lt;/i&gt;(-4)  (&lt;i&gt;i. e&lt;/i&gt;., adding -4  to the product of any two different components of them is a square). Similarly, each triple in the sequences  A, B, C and  D  has the property  &lt;i&gt;D&lt;/i&gt;(20). We show some interesting properties of these sequences that give various methods how to get squares from them.
%K Fibonacci Numbers
%K Lucas Numbers
%K Square
%K Symmetric Sum
%K Alternating Sum
%K Product
%K Component
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=7290