{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T09:05:28Z","timestamp":1767949528235,"version":"3.49.0"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T00:00:00Z","timestamp":1730764800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,5]],"date-time":"2024-11-05T00:00:00Z","timestamp":1730764800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s00200-024-00672-4","type":"journal-article","created":{"date-parts":[[2024,11,4]],"date-time":"2024-11-04T23:33:08Z","timestamp":1730763188000},"page":"131-147","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The complex-type k-Padovan sequences and their applications"],"prefix":"10.1007","volume":"37","author":[{"given":"\u00d6m\u00fcr","family":"Deveci","sequence":"first","affiliation":[]},{"given":"Anthony G.","family":"Shannon","sequence":"additional","affiliation":[]},{"given":"\u00d6zg\u00fcr","family":"Erda\u011f","sequence":"additional","affiliation":[]},{"given":"G\u00fcnta\u00e7","family":"Ceco","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,11,5]]},"reference":[{"issue":"1","key":"672_CR1","volume":"2014","author":"M Adam","year":"2014","unstructured":"Adam, M., Assimakis, N.: $$k$$-Step sum and $$m$$-step gap Fibonacci sequence. Int. Scho. Res. Not. 2014(1), 374902 (2014)","journal-title":"Int. Scho. Res. Not."},{"issue":"3","key":"672_CR2","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1080\/00150517.1998.12428931","volume":"36","author":"H Ayd\u0131n","year":"1998","unstructured":"Ayd\u0131n, H., Dikici, R.: General Fibonacci sequences in finite groups. Fibonacci Quart. 36(3), 216\u2013221 (1998)","journal-title":"Fibonacci Quart."},{"issue":"5","key":"672_CR3","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1007\/s00009-023-02467-2","volume":"20","author":"EF Bravo","year":"2023","unstructured":"Bravo, E.F.: Common values of Padovan and Perrin sequences. Mediterr. J. Math. 20(5), 268 (2023)","journal-title":"Mediterr. J. Math."},{"issue":"4","key":"672_CR4","first-page":"977","volume":"37","author":"JJ Bravo","year":"2022","unstructured":"Bravo, J.J., Herrera, J.L.: Generalized Padovan sequences. Commun. Korean Math. Soc. 37(4), 977\u2013988 (2022)","journal-title":"Commun. Korean Math. Soc."},{"key":"672_CR5","doi-asserted-by":"crossref","unstructured":"Campbell, C.M., Campbell, P.P.: On the Fibonacci length of powers of dihedral groups. In: Applications of Fibonacci numbers, Howard F.T (ed), Kluwer Academic Publisher, Dordrecht, vol. 9, pp. 69-85(2004)","DOI":"10.1007\/978-0-306-48517-6_9"},{"key":"672_CR6","doi-asserted-by":"crossref","unstructured":"Campbell, C.M., Doostie, H., Robertson, E.F.: Fibonacci length of generating pairs in groups. In: Bergum, G.E. (ed.) Applications of Fibonacci numbers. Springer, vol. 3, pp. 27\u201335. Kluwer Academic Publishers, Dordrecht, (1990)","DOI":"10.1007\/978-94-009-1910-5_4"},{"issue":"4","key":"672_CR7","first-page":"579","volume":"46","author":"\u00d6 Deveci","year":"2017","unstructured":"Deveci, \u00d6., Karaduman, E.: On the Padovan $$p$$-numbers. Hacet. J. Math. Stat. 46(4), 579\u2013592 (2017)","journal-title":"Hacet. J. Math. Stat."},{"key":"672_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2021.111522","volume":"153","author":"\u00d6 Deveci","year":"2021","unstructured":"Deveci, \u00d6., Hulku, S., Shannon, A.G.: On the co-complex-type $$k$$-Fibonacci numbers. Chaos Solitons Fractals. 153, 111522 (2021)","journal-title":"Chaos Solitons Fractals."},{"issue":"3","key":"672_CR9","doi-asserted-by":"publisher","first-page":"1352","DOI":"10.1080\/00927872.2020.1834573","volume":"49","author":"\u00d6 Deveci","year":"2021","unstructured":"Deveci, \u00d6., Shannon, A.G.: The complex-type $$k$$-Fibonacci sequences and their applications. Commun. Algebra. 49(3), 1352\u20131367 (2021)","journal-title":"Commun. Algebra."},{"issue":"12","key":"672_CR10","doi-asserted-by":"publisher","first-page":"5403","DOI":"10.1080\/00927872.2018.1468906","volume":"46","author":"\u00d6 Deveci","year":"2018","unstructured":"Deveci, \u00d6., Shannon, A.G.: The quaternion-Pell sequence. Commun. Algebra. 46(12), 5403\u20135409 (2018)","journal-title":"Commun. Algebra."},{"key":"672_CR11","unstructured":"Deveci, \u00d6., Gulliver, A., Hulku S., Apaydin, S.: Application of $$\\left( r,k\\right)$$-bonacci sequences (Submitted)"},{"issue":"1","key":"672_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.7546\/nntdm.2024.30.1.1-7","volume":"30","author":"O Di\u015fkaya","year":"2024","unstructured":"Di\u015fkaya, O., Menken, H.: On the pulsating Padovan sequence. Notes Number Theor Discret. Math. 30(1), 1\u20137 (2024)","journal-title":"Notes Number Theor Discret. Math."},{"issue":"1\u20132","key":"672_CR13","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/BF02831931","volume":"20","author":"H Doostie","year":"2006","unstructured":"Doostie, H., Hashemi, M.: Fibonacci lengths involving the Wall number $$K(n)$$. J. Appl. Math. Comput. 20(1\u20132), 171\u2013180 (2006)","journal-title":"J. Appl. Math. Comput."},{"key":"672_CR14","first-page":"71","volume-title":"Abstract algebra","author":"DS Dummit","year":"2004","unstructured":"Dummit, D.S., Foote, R.: Abstract algebra, 3rd edn., pp. 71\u201372. Wiley, New Jersey (2004)","edition":"3"},{"issue":"1","key":"672_CR15","doi-asserted-by":"publisher","first-page":"77","DOI":"10.5937\/MatMor2201077E","volume":"26","author":"\u00d6 Erda\u011f","year":"2022","unstructured":"Erda\u011f, \u00d6., Hal\u0131c\u0131, S., Deveci, \u00d6.: The complex-type Padovan-p sequences. Math. Moravica. 26(1), 77\u201388 (2022)","journal-title":"Math. Moravica."},{"issue":"1","key":"672_CR16","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1016\/j.chaos.2008.02.014","volume":"41","author":"S Falcon","year":"2009","unstructured":"Falcon, S., Plaza, A.: $$k$$-Fibonacci sequences modulo $$m$$. Chaos Solitons Fractals. 41(1), 497\u2013504 (2009)","journal-title":"Chaos Solitons Fractals."},{"key":"672_CR17","unstructured":"Gorenstein, D.: Finite groups. AMS Chelsea Publishing, pp. 188\u2013195 (1980)"},{"issue":"4","key":"672_CR18","first-page":"389","volume":"20","author":"M Hashemi","year":"2018","unstructured":"Hashemi, M., Pirzadeh, M.: $$t$$-nacci Sequences in some special groups of order $$m3$$. Mah. Rep. 20(4), 389\u2013399 (2018)","journal-title":"Mah. Rep."},{"issue":"3","key":"672_CR19","doi-asserted-by":"publisher","first-page":"289","DOI":"10.2307\/2313129","volume":"70","author":"AF Horadam","year":"1963","unstructured":"Horadam, A.F.: Complex Fibonacci numbers and Fibonacci quaternions. Amer. Math. Monthly. 70(3), 289\u2013291 (1963)","journal-title":"Amer. Math. Monthly."},{"key":"672_CR20","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1007\/978-3-642-64981-3","volume-title":"Endliche Gruppen","author":"B Huppert","year":"1967","unstructured":"Huppert, B.: Endliche Gruppen, pp. 90\u201393. Springer, New York (1967)"},{"issue":"1","key":"672_CR21","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1080\/00150517.1982.12430034","volume":"20","author":"D Kalman","year":"1982","unstructured":"Kalman, D.: Generalized Fibonacci numbers by matrix methods. Fibonacci Quart. 20(1), 73\u201376 (1982)","journal-title":"Fibonacci Quart."},{"issue":"1\u20132","key":"672_CR22","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/j.mcm.2009.03.010","volume":"50","author":"E Karaduman","year":"2009","unstructured":"Karaduman, E., Ayd\u0131n, H.: $$k$$-nacci sequences in some special groups of finite order. Math. Comput. Modell. 50(1\u20132), 53\u201358 (2009)","journal-title":"Math. Comput. Modell."},{"issue":"6","key":"672_CR23","doi-asserted-by":"publisher","first-page":"1915","DOI":"10.1216\/rmjm\/1181069352","volume":"36","author":"E K\u0131l\u0131\u00e7","year":"2006","unstructured":"K\u0131l\u0131\u00e7, E., Ta\u015f\u00e7\u0131, D.: On the generalized order- $$k$$ Fibonacci and Lucas numbers. Rocky Mt. J. Math. 36(6), 1915\u20131926 (2006)","journal-title":"Rocky Mt. J. Math."},{"issue":"2","key":"672_CR24","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1080\/00150517.1992.12429362","volume":"30","author":"SW Knox","year":"1992","unstructured":"Knox, S.W.: Fibonacci sequences in finite groups. Fibonacci Quart. 30(2), 116\u2013120 (1992)","journal-title":"Fibonacci Quart."},{"issue":"4","key":"672_CR25","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1007\/s40314-023-02331-9","volume":"42","author":"C K\u00f6me","year":"2023","unstructured":"K\u00f6me, C.: Factorizations and eigenvalues of the $$(r, k)$$ -bonacci matrices. Comput. Appl. Math. 42(4), 185 (2023)","journal-title":"Comput. Appl. Math."},{"key":"672_CR26","volume-title":"The theory of matrices: with applications","author":"P Lancaster","year":"1985","unstructured":"Lancaster, P., Tismenetsky, M.: The theory of matrices: with applications. Elsevier, Amsterdam (1985)"},{"issue":"2","key":"672_CR27","doi-asserted-by":"publisher","first-page":"266","DOI":"10.3390\/sym15020266","volume":"15","author":"G Lee","year":"2023","unstructured":"Lee, G., Kim, J.: On the Padovan codes and the Padovan cubes. Symmetry 15(2), 266 (2023)","journal-title":"Symmetry"},{"key":"672_CR28","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139172769","volume-title":"Introduction to finite fields and their applications","author":"R Lidl","year":"1994","unstructured":"Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications. Cambridge U.P, Cambridge (1994)"},{"issue":"1","key":"672_CR29","doi-asserted-by":"publisher","first-page":"1023123","DOI":"10.1080\/23311835.2015.1023123","volume":"2","author":"V Iliopoulos","year":"2015","unstructured":"Iliopoulos, V.: The plastic number and its generalized polynomial. Cogent Math. 2(1), 1023123 (2015)","journal-title":"Cogent Math."},{"key":"672_CR30","first-page":"169","volume":"71","author":"K Lu","year":"2006","unstructured":"Lu, K., Wang, J.: $$k$$-step Fibonacci sequence modulo $$m$$. Util. Math. 71, 169\u2013177 (2006)","journal-title":"Util. Math."},{"issue":"5","key":"672_CR31","doi-asserted-by":"publisher","first-page":"13537","DOI":"10.3934\/math.2024660","volume":"9","author":"E Mehraban","year":"2024","unstructured":"Mehraban, E., Gulliver, T.A., Boulaaras, S.M., Hosseini, K., Hincal, E.: New sequences from the generalized Pell $$p$$-numbers and mersenne numbers and their application in cryptography. AIMS Math. 9(5), 13537\u201313552 (2024)","journal-title":"AIMS Math."},{"key":"672_CR32","doi-asserted-by":"publisher","unstructured":"Mehraban, E., Gulliver, T.A., Hincal, E.: New cyclic groups based on the generalized order-$$k$$ Pell sequences in the Heisenberg group and their application in cryptography. Appl. Algebra Engrg. Comm. Comput, pp. 1\u201315 (2024). https:\/\/doi.org\/10.1007\/s00200-024-00649-3","DOI":"10.1007\/s00200-024-00649-3"},{"issue":"1","key":"672_CR33","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/S0096-3003(02)00360-0","volume":"143","author":"E \u00d6zkan","year":"2003","unstructured":"\u00d6zkan, E., Ayd\u0131n, H., Dikici, R.: $$3$$-step Fibonacci series modulo $$m$$. Appl. Math. Comput. 143(1), 165\u2013172 (2003)","journal-title":"Appl. Math. Comput."},{"key":"672_CR34","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1016\/j.amc.2014.01.014","volume":"232","author":"E \u00d6zkan","year":"2014","unstructured":"\u00d6zkan, E.: Truncated Lucas sequences and its period. Appl. Math. Comput. 232, 285\u2013291 (2014)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"672_CR35","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1080\/00150517.1974.12430736","volume":"12","author":"AG Shannon","year":"1974","unstructured":"Shannon, A.G.: Explicit expressions for powers of arbitrary order linear recursive sequences. Fibonacci Quart. 12(3), 281\u2013287 (1974)","journal-title":"Fibonacci Quart."},{"issue":"4","key":"672_CR36","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1080\/00150517.1974.12430699","volume":"12","author":"AG Shannon","year":"1974","unstructured":"Shannon, A.G.: Some properties of a fundamental recursive sequence of arbitrary order. Fibonacci Quart. 12(4), 327\u2013335 (1974)","journal-title":"Fibonacci Quart."},{"issue":"2","key":"672_CR37","first-page":"456","volume":"41","author":"S Tas","year":"2014","unstructured":"Tas, S., Karaduman, E.: The Padovan sequences in finite groups. Chiang Mai J. Sci. 41(2), 456\u2013462 (2014)","journal-title":"Chiang Mai J. Sci."},{"issue":"9\u201310","key":"672_CR38","doi-asserted-by":"publisher","first-page":"1763","DOI":"10.1016\/j.mcm.2010.07.003","volume":"52","author":"D Tasci","year":"2010","unstructured":"Tasci, D., Firengiz, M.C.: Incomplete Fibonacci and Lucas $$p$$ -numbers. Math. Comput. Modell. 52(9\u201310), 1763\u20131770 (2010)","journal-title":"Math. Comput. Modell."},{"issue":"6","key":"672_CR39","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1080\/00029890.1960.11989541","volume":"67","author":"DD Wall","year":"1960","unstructured":"Wall, D.D.: Fibonacci series modulo $$m$$. Amer. Math. Mon. 67(6), 525\u2013532 (1960)","journal-title":"Amer. Math. Mon."},{"issue":"4","key":"672_CR40","doi-asserted-by":"publisher","first-page":"356","DOI":"10.1080\/00150517.1986.12429747","volume":"24","author":"HJ Wilcox","year":"1986","unstructured":"Wilcox, H.J.: Fibonacci sequences of period $$n$$ in groups. Fibonacci Quart. 24(4), 356\u2013361 (1986)","journal-title":"Fibonacci Quart."}],"container-title":["Applicable Algebra in Engineering, Communication and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-024-00672-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00200-024-00672-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-024-00672-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T07:14:37Z","timestamp":1767942877000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00200-024-00672-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,5]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["672"],"URL":"https:\/\/doi.org\/10.1007\/s00200-024-00672-4","relation":{},"ISSN":["0938-1279","1432-0622"],"issn-type":[{"value":"0938-1279","type":"print"},{"value":"1432-0622","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,5]]},"assertion":[{"value":"18 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 October 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 November 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}