{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T04:09:55Z","timestamp":1748491795916,"version":"3.41.0"},"reference-count":8,"publisher":"Wiley","license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:p>We study the sum<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:msubsup><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msubsup><mml:mo>=<\/mml:mo><mml:mrow><mml:msubsup><mml:mo stretchy=\"false\">\u2211<\/mml:mo><mml:mrow><mml:mi>i<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msubsup><mml:mrow><mml:msub><mml:mrow><mml:mi>T<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><mml:mi>i<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mrow><\/mml:math>of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:math>step apart Tribonacci numbers for any<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mn>1<\/mml:mn><mml:mo>\u2264<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo>\u2264<\/mml:mo><mml:mi>k<\/mml:mi><\/mml:math>. We prove that<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>satisfies certain Tribonacci rule<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:msubsup><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msubsup><mml:mo>=<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:msubsup><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:msubsup><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>t<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:mi>\u03bb<\/mml:mi><\/mml:math>with integers<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math>, and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>\u03bb<\/mml:mi><\/mml:mrow><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2015\/301814","type":"journal-article","created":{"date-parts":[[2015,6,15]],"date-time":"2015-06-15T21:03:24Z","timestamp":1434402204000},"page":"1-8","source":"Crossref","is-referenced-by-count":1,"title":["On Partial Sum of Tribonacci Numbers"],"prefix":"10.1155","volume":"2015","author":[{"given":"Eunmi","family":"Choi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea"}]},{"given":"Jiin","family":"Jo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea"}]}],"member":"311","reference":[{"issue":"7","key":"5","volume":"16","year":"2013","journal-title":"Journal of Integer Sequences"},{"issue":"2","key":"6","first-page":"153","volume":"46","year":"2008","journal-title":"The Fibonacci Quarterly"},{"issue":"6","key":"8","first-page":"1030","volume":"31","year":"2011","journal-title":"Journal of Mathematical Research and Exposition"},{"key":"2","first-page":"73","volume":"100","year":"2011","journal-title":"Ars Combinatoria"},{"issue":"1","key":"3","doi-asserted-by":"crossref","first-page":"125","DOI":"10.18514\/MMN.2013.523","volume":"14","year":"2013","journal-title":"Miskolc Mathematical Notes"},{"key":"4","first-page":"13","volume":"86","year":"2008","journal-title":"Ars Combinatoria"},{"issue":"2","key":"7","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1080\/00150517.1988.12429644","volume":"26","year":"1988","journal-title":"Fibonacci Quarterly"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.14403\/jcms.2015.28.1.39"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2015\/301814.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2015\/301814.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2015\/301814.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T17:12:29Z","timestamp":1748452349000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2015\/301814\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"references-count":8,"alternative-id":["301814","301814"],"URL":"https:\/\/doi.org\/10.1155\/2015\/301814","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2015]]}}}