{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:39:43Z","timestamp":1753886383443,"version":"3.41.2"},"reference-count":16,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,2,26]],"date-time":"2012-02-26T00:00:00Z","timestamp":1330214400000},"content-version":"vor","delay-in-days":56,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100009391","name":"Kafkas University","doi-asserted-by":"crossref","award":["2010-FEF-61"],"award-info":[{"award-number":["2010-FEF-61"]}],"id":[{"id":"10.13039\/501100009391","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>We study the generalized order\u2010<jats:italic>k<\/jats:italic>Lucas sequences modulo<jats:italic>m<\/jats:italic>. Also, we define the<jats:italic>i<\/jats:italic>th generalized order\u2010<jats:italic>k<\/jats:italic>Lucas orbit (<jats:italic>G<\/jats:italic>) with respect to the generating set<jats:italic>A<\/jats:italic>and the constants<jats:italic>\u03b1<\/jats:italic><jats:sub>1<\/jats:sub>,<jats:italic>\u03b1<\/jats:italic><jats:sub>2<\/jats:sub>, and<jats:italic>\u03b1<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>\u22121<\/jats:sub>for a finite group<jats:italic>G<\/jats:italic>= \u2329<jats:italic>A<\/jats:italic>\u232a. Then, we obtain the lengths of the periods of the<jats:italic>i<\/jats:italic>th generalized order\u2010<jats:italic>k<\/jats:italic>Lucas orbits of the binary polyhedral groups \u2329<jats:italic>n<\/jats:italic>, 2, 2\u232a,\u23292,<jats:italic>n<\/jats:italic>, 2\u232a,\u23292, 2,<jats:italic>n<\/jats:italic>\u232a and the polyhedral groups (<jats:italic>n<\/jats:italic>, 2, 2), (2,<jats:italic>n<\/jats:italic>, 2), (2, 2,<jats:italic>n<\/jats:italic>) for 1 \u2264<jats:italic>i<\/jats:italic>\u2264<jats:italic>k<\/jats:italic>.<\/jats:p>","DOI":"10.1155\/2012\/464580","type":"journal-article","created":{"date-parts":[[2012,2,26]],"date-time":"2012-02-26T21:29:05Z","timestamp":1330291745000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Generalized Order\u2010<i>k<\/i>Lucas Sequences in Finite Groups"],"prefix":"10.1155","volume":"2012","author":[{"given":"\u00d6m\u00fcr","family":"Deveci","sequence":"first","affiliation":[]},{"given":"Erdal","family":"Karaduman","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,2,26]]},"reference":[{"key":"e_1_2_4_1_2","first-page":"169","article-title":"k-step Fibonacci sequence modulo m","volume":"71","author":"L\u00fc K.","year":"2006","journal-title":"Utilitas Mathematica"},{"key":"e_1_2_4_2_2","first-page":"193","article-title":"The periods of k-nacci sequences in centro-polyhedral groups and related groups","volume":"97","author":"Deveci \u00d6.","year":"2010","journal-title":"Ars Combinatoria"},{"key":"e_1_2_4_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(03)00804-X"},{"key":"e_1_2_4_4_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-05-08220-1"},{"key":"e_1_2_4_5_2","first-page":"21","article-title":"A class of Z-metacyclic groups involving the Lucas numbers","volume":"39","author":"Doostie H.","year":"2009","journal-title":"Novi Sad Journal of Mathematics"},{"key":"e_1_2_4_6_2","doi-asserted-by":"publisher","DOI":"10.1216\/rmjm\/1181069352"},{"key":"e_1_2_4_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(00)00204-4"},{"key":"e_1_2_4_8_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1993-1192974-3"},{"key":"e_1_2_4_9_2","doi-asserted-by":"publisher","DOI":"10.4134\/JKMS.2005.42.1.135"},{"key":"e_1_2_4_10_2","doi-asserted-by":"publisher","DOI":"10.2307\/2309169"},{"key":"e_1_2_4_11_2","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1080\/00150517.1992.12429362","article-title":"Fibonacci sequences in finite groups","volume":"30","author":"Knox S. 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