{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T01:27:15Z","timestamp":1768008435398,"version":"3.49.0"},"reference-count":13,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2016,12,1]],"date-time":"2016-12-01T00:00:00Z","timestamp":1480550400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>While discussing the sum of consecutive powers as a result of division of two binomials W.W. Sawyer [12] observes\n<jats:disp-quote>\n                     <jats:p>\u201cIt is a curious fact that most algebra textbooks give our ast result twice. It appears in two different chapters and usually there is no mention in either of these that it also occurs in the other. The first chapter, of course, is that on factors. The second is that on geometrical progressions. Geometrical progressions are involved in nearly all financial questions involving compound interest \u2013 mortgages, annuities, etc.\u201d<\/jats:p>\n                  <\/jats:disp-quote>\nIt\u2019s worth noticing that the first issue involves a simple arithmetical division of 99...9 by 9. While the above notion seems not have changed over the last 50 years, it reflects only a special case of a broader class of problems involving two variables. It seems strange, that while binomial formula is discussed and studied widely [7], [8], little research is done on its counterpart with all coefficients equal to one, which we will call here the subnomial. The study focuses on its basic properties and applies it to some simple problems usually proven by induction [6].<\/jats:p>","DOI":"10.1515\/forma-2016-0022","type":"journal-article","created":{"date-parts":[[2017,2,25]],"date-time":"2017-02-25T10:00:53Z","timestamp":1488016853000},"page":"261-273","source":"Crossref","is-referenced-by-count":3,"title":["On Subnomials"],"prefix":"10.1515","volume":"24","author":[{"given":"Rafa\u0142","family":"Ziobro","sequence":"first","affiliation":[{"name":"Department of Carbohydrate Technology, University of Agriculture, Krakow, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,2,23]]},"reference":[{"key":"2021040802442472642_j_forma-2016-0022_ref_001_w2aab2b8b8b1b7b1ab1ab1Aa","unstructured":"[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41\u201346, 1990."},{"key":"2021040802442472642_j_forma-2016-0022_ref_002_w2aab2b8b8b1b7b1ab1ab2Aa","unstructured":"[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107\u2013114, 1990."},{"key":"2021040802442472642_j_forma-2016-0022_ref_003_w2aab2b8b8b1b7b1ab1ab3Aa","unstructured":"[3] Czes\u0142aw Byli\u0144ski. Some properties of restrictions of finite sequences. Formalized Mathematics, 5(2):241\u2013245, 1996."},{"key":"2021040802442472642_j_forma-2016-0022_ref_004_w2aab2b8b8b1b7b1ab1ab4Aa","unstructured":"[4] Czes\u0142aw Byli\u0144ski. Functions and their basic properties. Formalized Mathematics, 1(1): 55\u201365, 1990."},{"key":"2021040802442472642_j_forma-2016-0022_ref_005_w2aab2b8b8b1b7b1ab1ab5Aa","unstructured":"[5] Wenpai Chang, Hiroshi Yamazaki, and Yatsuka Nakamura. The inner product and conjugate of finite sequences of complex numbers. Formalized Mathematics, 13(3):367\u2013373, 2005."},{"key":"2021040802442472642_j_forma-2016-0022_ref_006_w2aab2b8b8b1b7b1ab1ab6Aa","unstructured":"[6] Jacek Gancarzewicz. Arytmetyka. Wydawnictwo UJ, Krak\u00f3w, 2000. In Polish."},{"key":"2021040802442472642_j_forma-2016-0022_ref_007_w2aab2b8b8b1b7b1ab1ab7Aa","doi-asserted-by":"crossref","unstructured":"[7] Cao Hui-Qin and Pan Hao. Factors of alternating binomial sums. Advances in Applied Mathematics, 45(1):96 \u2013 107, 2010. doi:http:\/\/dx.doi.org\/10.1016\/j.aam.2009.09.004.","DOI":"10.1016\/j.aam.2009.09.004"},{"key":"2021040802442472642_j_forma-2016-0022_ref_008_w2aab2b8b8b1b7b1ab1ab8Aa","doi-asserted-by":"crossref","unstructured":"[8] Sudesh K. Khanduja, Ramneek Khassa, and Shanta Laishram. Some irreducibility results for truncated binomial expansions. Journal of Number Theory, 131(2):300 \u2013 308, 2011. doi:http:\/\/dx.doi.org\/10.1016\/j.jnt.2010.08.004.","DOI":"10.1016\/j.jnt.2010.08.004"},{"key":"2021040802442472642_j_forma-2016-0022_ref_009_w2aab2b8b8b1b7b1ab1ab9Aa","unstructured":"[9] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4): 573\u2013577, 1997."},{"key":"2021040802442472642_j_forma-2016-0022_ref_010_w2aab2b8b8b1b7b1ab1ac10Aa","unstructured":"[10] Rafa\u0142 Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887\u2013890, 1990."},{"key":"2021040802442472642_j_forma-2016-0022_ref_011_w2aab2b8b8b1b7b1ab1ac11Aa","unstructured":"[11] Rafa\u0142 Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829\u2013832, 1990."},{"key":"2021040802442472642_j_forma-2016-0022_ref_012_w2aab2b8b8b1b7b1ab1ac12Aa","unstructured":"[12] W.W. Sawyer. The Search for Pattern. Penguin Books Ltd, Harmondsworth, Middlessex, England, 1970."},{"key":"2021040802442472642_j_forma-2016-0022_ref_013_w2aab2b8b8b1b7b1ab1ac13Aa","unstructured":"[13] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569\u2013573, 1990."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/24\/4\/article-p261.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2016-0022","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,8]],"date-time":"2021-04-08T11:24:22Z","timestamp":1617881062000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2016-0022"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,1]]},"references-count":13,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,2,23]]},"published-print":{"date-parts":[[2016,12,1]]}},"alternative-id":["10.1515\/forma-2016-0022"],"URL":"https:\/\/doi.org\/10.1515\/forma-2016-0022","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2016,12,1]]}}}