{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T13:27:58Z","timestamp":1777037278868,"version":"3.51.4"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,17]],"date-time":"2021-01-17T00:00:00Z","timestamp":1610841600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771351"],"award-info":[{"award-number":["11771351"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Northwest University Doctorate Dissertation of Excellence Funds","award":["YYB17002"],"award-info":[{"award-number":["YYB17002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider the recurrence properties of two generalized forms of Narayana\u2019s cows sequence. On the one hand, we study Narayana\u2019s cows sequence at negative indices and express it as the linear combination of the sequence at positive indices. On the other hand, we study the convolved Narayana number and obtain a computation formula for it.<\/jats:p>","DOI":"10.3390\/sym13010149","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T02:36:05Z","timestamp":1611196565000},"page":"149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On the Recurrence Properties of Narayana\u2019s Cows Sequence"],"prefix":"10.3390","volume":"13","author":[{"given":"Xin","family":"Lin","sequence":"first","affiliation":[{"name":"School of Mathematics, Northwest University, Xi\u2019an 710127, Shaanxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,17]]},"reference":[{"key":"ref_1","unstructured":"Allouche, J.P., and Johnson, T. (1996). Narayana\u2019s cows and delayed morphisms. J. d\u2019Inform. Music., hal-02986050. 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Math."},{"key":"ref_12","first-page":"195","article-title":"Generating Functions K-Fibonacci and K-Jacobsthal Numbers at Negative Indices","volume":"6","author":"Boussayoud","year":"2018","journal-title":"Electron. J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"121","DOI":"10.12691\/tjant-5-4-1","article-title":"On the k-Lucas Numbers and Lucas Polynomials","volume":"5","author":"Boussayoud","year":"2017","journal-title":"Turk. J. Anal. Number Theory"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Soykan, Y. (2019). Summing Formulas For Generalized Tribonacci Numbers. arXiv.","DOI":"10.20944\/preprints201910.0172.v1"},{"key":"ref_15","unstructured":"Koshy, T. (2011). 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