{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T19:08:51Z","timestamp":1760382531445},"reference-count":13,"publisher":"World Scientific Pub Co Pte Lt","issue":"03n04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[1999,6]]},"abstract":"<jats:p> Every positive integer can be written as a sum of Fibonacci numbers; it can also be written as a (finite) sum of (positive and negative) powers of the golden mean \u03c6. We show that there exists a letter-to-letter finite two-tape automaton that maps the Fibonacci representation of any positive integer onto its \u03c6-expansion, provided the latter is folded around the radix point. As a corollary, the set of \u03c6-expansions of the positive integers is a linear context-free language. These results are actually proved in the more general case of quadratic Pisot units. <\/jats:p><jats:p> <\/jats:p><jats:p> R\u00e9sum\u00e9: Tout nombre entier positif peut s'\u00e9crire comme une somme de nombres de Fibonacci; tout entier peut \u00e9galement s'\u00e9crire comme une somme (finie) de puissances (positives et n\u00e9gatives) du \"nombre d'or\" \u03c6. Nous montrons qu'il existe un automate \u00e0 deux bandes, fini et lettre-\u00e0-lettre, qui envoie la repr\u00e9sentation d'un entier en base de Fibonacci sur sa repr\u00e9sentation dans la base \u03c6 modulo le fait qu'on a repli\u00e9 cette derni\u00e8re autour du point d\u00e9cimal. On en d\u00e9duit que l'ensemble des repr\u00e9sentations des entiers en base \u03c6 est un langage context-free lin\u00e9aire. Tous ces r\u00e9sultats sont en fait \u00e9tablis dans le cas g\u00e9n\u00e9ral o\u00f9 la base consid\u00e9r\u00e9e est un nombre de Pisot quadratique unitaire. <\/jats:p>","DOI":"10.1142\/s0218196799000230","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:05:54Z","timestamp":1027767954000},"page":"351-384","source":"Crossref","is-referenced-by-count":8,"title":["AUTOMATIC CONVERSION FROM FIBONACCI REPRESENTATION TO REPRESENTATION IN BASE \u03c6, AND A GENERALIZATION"],"prefix":"10.1142","volume":"09","author":[{"given":"CHRISTIANE","family":"FROUGNY","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris 8 and Laboratoire Informatique Algorithmique: Fondements et Applications, Universit\u00e9 Paris 7 and C.N.R.S., France"}]},{"given":"JACQUES","family":"SAKAROVITCH","sequence":"additional","affiliation":[{"name":"Laboratoire Traitement et Communication, de l'Information (C.N.R.S. URA 820), E.N.S.T., Paris, France"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"p_1","doi-asserted-by":"publisher","DOI":"10.1109\/TEC.1961.5219227"},{"key":"p_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01578846"},{"key":"p_4","first-page":"419","volume":"285","author":"A","year":"1977","journal-title":"C. R. Acad. Sci. Paris"},{"key":"p_5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01952053"},{"key":"p_7","doi-asserted-by":"publisher","DOI":"10.1147\/rd.91.0047"},{"key":"p_9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01368783"},{"key":"p_10","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(93)90230-Q"},{"key":"p_12","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700007057"},{"key":"p_17","doi-asserted-by":"publisher","DOI":"10.1007\/BF02020954"},{"key":"p_18","first-page":"1","author":"Perrin D.","year":"1990","journal-title":"Elsevier"},{"key":"p_19","doi-asserted-by":"publisher","DOI":"10.1007\/BF02020331"},{"key":"p_20","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(67)80006-8"},{"key":"p_21","first-page":"1","volume":"4","author":"Zeckendorf E.","year":"1972","journal-title":"Bull. Soc. Roy. Liege"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196799000230","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T21:56:00Z","timestamp":1565128560000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196799000230"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,6]]},"references-count":13,"journal-issue":{"issue":"03n04","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[1999,6]]}},"alternative-id":["10.1142\/S0218196799000230"],"URL":"https:\/\/doi.org\/10.1142\/s0218196799000230","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,6]]}}}