{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T21:07:07Z","timestamp":1765487227579,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T00:00:00Z","timestamp":1762128000000},"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":"crossref","award":["12071063"],"award-info":[{"award-number":["12071063"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The Horadam sequence {Hn(a,b;p,q)}n\u2a7e0 has been widely studied in combinatorics and number theory. In this paper, we find that the context-free grammar G={x\u2192px+y,y\u2192qx} can be used to generate Horadam sequences. Using this grammar, we deduce several identities, including Cassini-like identities. Moreover, we investigate the relationship between two distinct Horadam sequences Hn(a,b;p,q) and Hn(c,d;p,q) with (a,b)\u2260(c,d) and provide an approach to derive identities, which can be illustrated by the Fibonacci and Lucas sequences as well as the two kinds of Chebyshev polynomials.<\/jats:p>","DOI":"10.3390\/axioms14110819","type":"journal-article","created":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T19:30:27Z","timestamp":1762198227000},"page":"819","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Grammatical Interpretation of Horadam Sequences"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6661-445X","authenticated-orcid":false,"given":"Jun-Ying","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hai-Ling","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhi-Hong","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1080\/00029890.1961.11989696","article-title":"A generalized Fibonacci sequence","volume":"68","author":"Horadam","year":"1961","journal-title":"Am. 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