This paper presents an enumeration algorithm to generate all magic squares of order 5 based on the ideas of basic form (Schroeppel ) and generating vector which is extension of Frénicle Quads (Ollerenshaw and Bondi ). The results lead us to extend Frénicle-Amela patterns from the case of order 4 to the case of order 5, which we refer to Frénicle-Amela-Like patterns. We show that these interesting Frénicle-Amela-Like patterns appear simultaneously. The number of these patterns is also calculated.
 Candy, A. L., 1937: Construction, Classification and Census of Magic Squares of Order 5. Edwards brothers, 249 pp. Search in Google Scholar
 Chinese Magic Square, 2014: Accessed 12 June 2014. Search in Google Scholar
[Available online at http://www.zhghf.net/.] Search in Google Scholar
 Clifford, A. P., 2003: The Zen of Magic Squares, Circles, and Stars. Princeton University Press, 373 pp. Search in Google Scholar
 Fang, K. T., Luo, Y. Y., and Zheng, Y. X., 2015: Classification of magic squares of order 4. Proc. IWMS. Haikou, China, International Workshop on Matrices and Statistics, 84-97. Search in Google Scholar
 Garder, M., 1975: Mathematical games, A breakthrough in magic squares, and the first perfect magic cube. Scientific American, 118-123. Search in Google Scholar
 Ollerenshaw, K., and Bondi, H., 1982:Magic squares of order four. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 306, 443-532. 10.1098/rsta.1982.0093Search in Google Scholar
 Schroeppel, R., 1976: The Order 5 Magic Squares Program, Scientific American. Search in Google Scholar
 Styan, G. P. H., 2014: Some illustrated comments on 5 x 5 goldenmagicmatrices and on 5 x 5 Stifelsche Quadrate. 23rd Conf. International Workshop on Matrices and Statistics, Ljubljana, Slovenia, 41 pp. Search in Google Scholar
 Trump, W., 2012: How many magic squares are there? Accessed 12 June 2014. [Available online at http://www.trump.de/ magic-squares/howmany.html.] Search in Google Scholar
 Baidu Tieba, 2010: Accessed 11 June 2014. [Available online at http://tieba.baidu.com/p/957776994?pid=10675158083& cid=0&from=prin\sharp10675158083?from=prin.] Search in Google Scholar
©2016 Ziqi Lin et al.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.