{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T06:24:24Z","timestamp":1771050264805,"version":"3.50.1"},"publisher-location":"Cham","reference-count":31,"publisher":"Springer International Publishing","isbn-type":[{"value":"9783030964009","type":"print"},{"value":"9783030964016","type":"electronic"}],"license":[{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022]]},"DOI":"10.1007\/978-3-030-96401-6_24","type":"book-chapter","created":{"date-parts":[[2022,4,15]],"date-time":"2022-04-15T17:09:24Z","timestamp":1650042564000},"page":"259-269","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["From Fibonacci Sequence to More Recent Generalisations"],"prefix":"10.1007","author":[{"given":"Paula","family":"Catarino","sequence":"first","affiliation":[]},{"given":"Helena","family":"Campos","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,16]]},"reference":[{"key":"24_CR1","doi-asserted-by":"crossref","unstructured":"Dil, A., Mezo,I., A Symmetric algorithm hyperharmonic and Fibonacci Numbers. Appl. Math. Comp., (206): 942\u2013951, 2006.","DOI":"10.1016\/j.amc.2008.10.013"},{"key":"24_CR2","unstructured":"Hoggatt, V. E., Fibonacci and Lucas Numbers, A publication of the Fibonacci, Association. University of Santa Clara, Santa Clara. Houghton Mifflin Company, 1969."},{"key":"24_CR3","unstructured":"Vorobiov, N. N., N\u00fameros de Fibonacci, Editora MIR, URSS, 1974."},{"key":"24_CR4","unstructured":"Vajda, S., Fibonacci and Lucas numbers, and the Golden section, theory and applications. Ellis Horwood limited, 1989."},{"key":"24_CR5","doi-asserted-by":"crossref","unstructured":"Campos, H., Catarino, P., Aires, A. P., Vasco, P. and Borges, A., On Some Identities of $$k$$-Jacobsthal-Lucas Numbers, Int. Journal of Math. Analysis, 8(10): 489\u2013494, 2014.","DOI":"10.12988\/ijma.2014.4249"},{"key":"24_CR6","unstructured":"Catarino, P., Vasco, P. and Campos, H., A note on $$k$$-Pell, $$k$$-Pell-Lucas and modified $$k$$-Pell Numbers with arithmetic indexes, Acta Mathematica Universitatis Comenianae, 89(1): 97\u2013107, 2020."},{"key":"24_CR7","unstructured":"Catarino, P., Vasco, P., and Campos, H., On some identities for balancing and cobalancing numbers, Annales Mathematicae et Informaticae, 45: 11\u201324, 2015."},{"key":"24_CR8","unstructured":"Catarino, P. and Campos, H., Incomplete $$k$$-Pell, $$k$$-Pell-Lucas and modified $$k$$-Pell numbers, Hacettepe journal of mathematics and statistics, 46(3): 361\u2013372, 2017."},{"key":"24_CR9","unstructured":"Catarino, P., Morais, E. and Campos, H., A note on $$k$$-Telephone and Incomplete k-Telephone numbers, 2021. (to appear)"},{"key":"24_CR10","unstructured":"Catarino, P. and Borges, A., On Leonardo numbers, Acta Mathematica Universitatis Comenianae, 89(1): 75\u201386, 2019."},{"key":"24_CR11","unstructured":"Catarino, P. and Borges, A., A Note on Incomplete Leonardo numbers, Integers, 20, 2020."},{"key":"24_CR12","doi-asserted-by":"crossref","unstructured":"Vieira, R. P. M., Alves, F. R. V. and Catarino, P., Rela\u00e7\u00f5es Bidimensionais e Identidades da sequ\u00eancia de Leonardo, Revista Sergipana de Matem\u00e1tica e Educa\u00e7\u00e3o Matem\u00e1tica, 2: 156\u2013173, 2019.","DOI":"10.34179\/revisem.v4i2.11863"},{"key":"24_CR13","unstructured":"Catarino, P., Alves, F. R. V. and Campos, H., On Hyper $$k$$-Pell, Hyper $$k$$-Pell-Lucas and Hyper Modified $$k$$-Pell sequences, 2021. (to appear)"},{"key":"24_CR14","unstructured":"Horadam, A. F., Basic properties of a certain generalized sequence of numbers. Fibonacci Quart. 3(3): 161\u2013176, 1965."},{"key":"24_CR15","doi-asserted-by":"crossref","unstructured":"Horadam, A. F., Generating functions for powers of a certain generalized sequences of numbers. Duke Math. J. 32(3): 437\u2013446, 1965.","DOI":"10.1215\/S0012-7094-65-03244-8"},{"key":"24_CR16","doi-asserted-by":"crossref","unstructured":"Catarino, P., On Some Identities and Generating Functions for $$k$$-Pell Numbers. Int. J. Math. Anal. (Ruse), 7(38): 1877\u20131884, 2013.","DOI":"10.12988\/ijma.2013.35131"},{"key":"24_CR17","doi-asserted-by":"crossref","unstructured":"Catarino, P. and Vasco, P. On some Identities and Generating Functions for $$k$$-Pell-Lucas sequence. Appl. Math. Sci, 7(98): 4867\u20134873, 2013.","DOI":"10.12988\/ams.2013.37379"},{"key":"24_CR18","doi-asserted-by":"crossref","unstructured":"Catarino, P. and Vasco, P. Modified $$k$$-Pell Sequence: Some Identities and Ordinary Generating Function. Appl. Math. Sci, 7(121): 6031\u20136037, 2013.","DOI":"10.12988\/ams.2013.39513"},{"key":"24_CR19","doi-asserted-by":"crossref","unstructured":"Catarino, P. and Vasco, P. Some basic properties and a two-by-two matrix involving the $$k$$-Pell Numbers. Int. J. Math. Anal. (Ruse), 7(45): 2209\u20132215, 2013.","DOI":"10.12988\/ijma.2013.37172"},{"key":"24_CR20","doi-asserted-by":"crossref","unstructured":"Vasco, P. and Catarino, P. Sums and products involving terms of $$k$$-Pell, $$k$$-Pell-Lucas and Modified $$k$$-Pell sequences, JP Journal of Algebra, Number Theory and Applications, 32(2): 87\u201398, 2014.","DOI":"10.17654\/JPANTAMar2014_087_098"},{"key":"24_CR21","doi-asserted-by":"crossref","unstructured":"Vasco, P., Catarino, P., Campos,H., Aires, A. P. and Borges, A., $$k$$-Pell, $$k$$-Pell-Lucas and modified $$k$$-Pell numbers: Some identities and norms of Hankel matrices, International Journal of Mathematical Analysis, 9(1): 31\u201337, 2015.","DOI":"10.12988\/ijma.2015.411370"},{"key":"24_CR22","unstructured":"Wani, A. A., Badshah, V.H., Rathore,G. B. S. and Catarino, P., Generalized Fibonacci and $$k$$-Pell matrix sequences, Punjab University Journal of Mathematics, 50(1): 17\u201328, 2019."},{"key":"24_CR23","unstructured":"Wani, A. A., Catarino,P. and Halici, S. On a Study of $$(s, t)$$-Generalized Pell Sequence and its Matrix Sequence, Punjab University Journal of Mathematics, 51(9): 17\u201332, 2019."},{"key":"24_CR24","doi-asserted-by":"crossref","unstructured":"Catarino, P. and Campos, H., A note on Gaussian Modified Pell numbers, Journal of Information and Optimization Sciences, 39(6): 1363\u20131371, 2018.","DOI":"10.1080\/02522667.2018.1471267"},{"key":"24_CR25","doi-asserted-by":"crossref","unstructured":"Catarino, P., On some identities for $$k$$-Fibonacci sequence, Int. J. Contemp. Math. Sciences, 9(1): 37\u201342, 2014.","DOI":"10.12988\/ijcms.2014.311120"},{"key":"24_CR26","doi-asserted-by":"crossref","unstructured":"Catarino, P., Vasco, P., Borges, A., Campos,H. and Aires, A. P., Sums, Products and identities involving k-Fibonacci and k-Lucas sequences, JP Journal of Algebra, Number Theory and Applications, 32(2): 63\u201377, 2014.","DOI":"10.17654\/JPANTAFeb2014_063_077"},{"key":"24_CR27","doi-asserted-by":"crossref","unstructured":"Borges, A., Catarino, P., Aires, A. P., Vasco, P. and Campos, H., Two-by-Two Matrices Involving $$k$$-Fibonacci and $$k$$-Lucas Sequences, Applied Mathematical Sciences, 34(8): 1659\u20131666, 2014.","DOI":"10.12988\/ams.2014.4280"},{"key":"24_CR28","unstructured":"Wani, A. A., Catarino, P. and Rafiq, R., On the Properties of $$k$$-Fibonacci-Like Sequence, Int. J. Math. And Appl., 6(1-A): 187\u2013198, 2018."},{"key":"24_CR29","unstructured":"Catarino, P., Vasco P., Campos, H., Aires, A. P. and Borges, A., New families of Jacobsthal and Jacobsthal-Lucas numbers. Algebra and Discrete Mathematics, 20(1): 40\u201354, 2015."},{"key":"24_CR30","unstructured":"Catarino, P., On Jacobsthal difference sequences, Acta Math. Univ. Comenianae, LXXXVII(2): 267\u2013276, 2018."},{"key":"24_CR31","unstructured":"Behera, A. and Panda, G. K., On the square roots of triangular numbers, Fibonacci Quart., 37(2): 98\u2013105, 1999."}],"container-title":["Springer Proceedings in Mathematics & Statistics","Mathematical Methods for Engineering Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-030-96401-6_24","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,15]],"date-time":"2022-04-15T17:16:35Z","timestamp":1650042995000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-030-96401-6_24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022]]},"ISBN":["9783030964009","9783030964016"],"references-count":31,"URL":"https:\/\/doi.org\/10.1007\/978-3-030-96401-6_24","relation":{},"ISSN":["2194-1009","2194-1017"],"issn-type":[{"value":"2194-1009","type":"print"},{"value":"2194-1017","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022]]},"assertion":[{"value":"16 April 2022","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"ICMASE","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Mathematics and its Applications in Science and Engineering","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2021","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"1 July 2021","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2 July 2021","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"icmase2021","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/2021.icmase.com\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}