{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:38:58Z","timestamp":1760143138944,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,25]],"date-time":"2024-01-25T00:00:00Z","timestamp":1706140800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Innovation and Technology of Hungary from the National Research, Development, and Innovation Fund","award":["TKP2021-NVA-09"],"award-info":[{"award-number":["TKP2021-NVA-09"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The binary number system is the basic number representation in computing. We can encode natural numbers with finite 0-1 sequences. The representation of natural numbers is based on this system. However, this poses problems and is technically not perfect. Several attempts have been made to handle integers (signed numbers). We mention only two: the balanced triple number system and the number system with base \u22122. Our paper introduces new possibilities. We also shed light on the graph theoretical background of the new number systems.<\/jats:p>","DOI":"10.3390\/a17020055","type":"journal-article","created":{"date-parts":[[2024,1,25]],"date-time":"2024-01-25T10:32:38Z","timestamp":1706178758000},"page":"55","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Binary Numeration System with Alternating Signed Digits and Its Graph Theoretical Relationship"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8487-233X","authenticated-orcid":false,"given":"P\u00e9ter","family":"Hajnal","sequence":"first","affiliation":[{"name":"Bolyai Institute, University of Szeged, Aradi v\u00e9rtan\u00fak tere 1, 6720 Szeged, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Allouche, J.-P., and Shallit, J. (2003). Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press. Chapter 3: Numeration Systems.","DOI":"10.1017\/CBO9780511546563"},{"key":"ref_2","first-page":"161","article-title":"A Short Account of Negativo-Affirmative Arithmetick","volume":"34","author":"Colson","year":"1726","journal-title":"Philos. Trans."},{"key":"ref_3","unstructured":"Cauchy, A. (1840). Comptes Rendus Hebdo Madaires des S\u00e9ances de l\u2019Acad\u00e9mie des Sciences, Elsevier."},{"key":"ref_4","first-page":"11","article-title":"Are there negative digits?","volume":"4","year":"2021","journal-title":"Kvant"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1080\/00029890.1950.11999490","article-title":"A Symmetrical Notation for Numbers","volume":"57","author":"Shannon","year":"1950","journal-title":"Am. Math. Mon."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1049\/ep.1964.0037","article-title":"The ternary computer","volume":"10","author":"Alexander","year":"1964","journal-title":"Electron. Power"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Eaton, M. (2012, January 23\u201324). Design and construction of a balanced ternary ALU with potential future cybernetic intelligent systems applications. Proceedings of the 2012 IEEE 11th International Conference on Cybernetic Intelligent Systems (CIS), Limerick, Ireland.","DOI":"10.1109\/CIS.2013.6782156"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/MAHC.2005.49","article-title":"The Ternary Calculating Machine of Thomas Fowler","volume":"27","author":"Glusker","year":"2005","journal-title":"Ann. Hist. Comput."},{"key":"ref_9","first-page":"1585","article-title":"Ternary arithmetic unit","volume":"115","author":"Halpern","year":"1968","journal-title":"Proc. IEEE"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1511\/2001.40.490","article-title":"Third Base","volume":"89","author":"Hayes","year":"2001","journal-title":"Am. Sci."},{"key":"ref_11","unstructured":"Hayes, B. (2008). Group Theory in the Bedroom, and Other Mathematical Diversions, Hill and Wang, A Division of Farrar, Straus and Giroux. Chapter 10."},{"key":"ref_12","unstructured":"Selling, E. (1887). Eine Neue Rechenmaschine, Springer."},{"key":"ref_13","unstructured":"Brousentsov, N.P., Maslov, S.P., Alvarez, J.R., and Zhogolev, E.A. (2024, January 22). Development of Ternary Computers at Moscow State University, Russian Virtual Computer Museum. Available online: https:\/\/www.computer-museum.ru\/english\/setun.htm."},{"key":"ref_14","unstructured":"Fowler, T. (2010). Tables for Facilitating Arithmetical Calculations Intended for Calculating the Proportionate Charges on the Parishes in Poor Law Unions, Nabu Press."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Frieder, G., and Luk, C. (1972, January 25\u201326). Ternary Computer. Proceedings of the 5th Annual Workshop on Microprogramming Elektronika, Urbana, IL, USA.","DOI":"10.1145\/776378.776393"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1109\/T-C.1975.224188","article-title":"Algorithms for binary coded balanced and ordinary ternary operations","volume":"24","author":"Frieder","year":"1975","journal-title":"IEEE Trans. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1038\/scientificamerican0564-118","article-title":"Mathematical Games, The \u201ctyranny of 10\u201d overthrown with the ternary number system","volume":"210","author":"Gardner","year":"1964","journal-title":"Sci. Am."},{"key":"ref_18","unstructured":"Knuth, D.E. (2011). The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Addison-Wesley. [3rd ed.]."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/0020-0190(87)90005-6","article-title":"Jump interpolation search trees and symmetric binary numbers","volume":"26","author":"Paul","year":"1987","journal-title":"Inf. Process. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Franklin, M. (2004). Advances in Cryptology\u2014CRYPTO 2004, Springer.","DOI":"10.1007\/b99099"},{"key":"ref_21","unstructured":"Prodinger, H. (2000). On binary representations of integers with digits. Integers, A8."},{"key":"ref_22","unstructured":"Shallit, J. (2024, January 22). A Primer on Balanced Binary Representations. Available online: https:\/\/cs.uwaterloo.ca\/~shallit\/Papers\/bbr.pdf."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Shallit, J., Shan, S.L., and Yang, K.H. (2022). Automatic Sequences in Negative Bases and Proofs of Some Conjectures of Shevelev. arXiv.","DOI":"10.1051\/ita\/2022011"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"228","DOI":"10.4204\/EPTCS.386.18","article-title":"A General Approach to Proving Properties of Fibonacci Representations via Automata Theory","volume":"386","author":"Shallit","year":"2023","journal-title":"Electron. Proc. Theor. Comput. Sci."},{"key":"ref_25","unstructured":"Zeckendorf, E. Unpublished observation from 1939."},{"key":"ref_26","first-page":"179","article-title":"Repr\u00e9sentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas","volume":"41","author":"Zeckendorf","year":"1972","journal-title":"Bull. Soc. Roy. Sci. Li\u00e8ge"},{"key":"ref_27","unstructured":"Stevin, S. (1951). Voorstelling van Natuurlijke Getallen Door een som van Getallen van Fibonacci, Stichting Mathematisch Centrum."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/BF02940581","article-title":"Bemerkungen zur Theorie der Diophantischen Approximationen","volume":"1","author":"Ostrowski","year":"1922","journal-title":"Abh. Math. Sem. Hamburg"},{"key":"ref_29","unstructured":"OEIS (2024, January 22). The On-Line Encyclopedia of Integer Sequences. Available online: https:\/\/oeis.org\/."},{"key":"ref_30","first-page":"745","article-title":"Differences of multiple Fibonacci numbers","volume":"9","author":"Alpert","year":"2009","journal-title":"Integers Electron. J. Comb. Number Theory"},{"key":"ref_31","first-page":"54","article-title":"A short note on Zeckendorf type numeration systems with negative digits allowed","volume":"97","author":"Hajnal","year":"2023","journal-title":"Bull. ICA"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1080\/00150517.2013.12427944","article-title":"Efficient algorithms for Zeckendorf arithmetic","volume":"51","author":"Ahlbach","year":"2013","journal-title":"Fibonacci Q."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1109\/TEC.1961.5219227","article-title":"Signed-Digit Number Representations for Fast Parallel Arithmetic","volume":"EC-10","author":"Avizienis","year":"1961","journal-title":"IRE Trans. Electron. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1093\/qjmam\/4.2.236","article-title":"A signed binary multiplication technique","volume":"4","author":"Booth","year":"1951","journal-title":"Quart. J. Mech. Appl. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.protcy.2012.05.043","article-title":"Algorithms for Ternary Number System","volume":"4","author":"Das","year":"2012","journal-title":"Procedia Technol."},{"key":"ref_36","unstructured":"Idziaszek, T. (June, January 30). Efficient Algorithm for Multiplication of Numbers in Zeckendorf Representation. Proceedings of the 10th International Conference on Fun with Algorithms (FUN 2021), Sicily, Italy. Available online: https:\/\/drops.dagstuhl.de\/opus\/volltexte\/2020\/12777."},{"key":"ref_37","first-page":"267","article-title":"Russian Peasant Multiplication and Egyptian Division in Zeckendorf Arithmetic","volume":"30","author":"Tee","year":"2003","journal-title":"Austral. Math. Soc. Gaz."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1080\/00150517.2014.12427898","article-title":"A Generalization of Fibonacci Far-Difference Representations and Gaussian Behavior","volume":"52","author":"Demontigny","year":"2017","journal-title":"Fibonacci Q."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1080\/00150517.1992.12429361","article-title":"Zeckendorf representations using negative Fibonacci numbers","volume":"30","author":"Bunder","year":"1992","journal-title":"Fibonacci Q."},{"key":"ref_40","unstructured":"Haran, B., and Sloane, N.J.A. (2024, January 22). Problems with Powers of Two, Numberphile Youtube Channel. September 2022. Available online: https:\/\/youtu.be\/IPoh5C9CcI8."},{"key":"ref_41","unstructured":"Scheuerle, T. (2024, January 22). Comment to A352178 in The On-Line Encyclopedia of Integer Sequences. 23 September 2022. Available online: https:\/\/oeis.org\/A352178."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/2\/55\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:49:22Z","timestamp":1760104162000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/2\/55"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,25]]},"references-count":41,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["a17020055"],"URL":"https:\/\/doi.org\/10.3390\/a17020055","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2024,1,25]]}}}