{"id":1194,"date":"2015-07-29T09:53:45","date_gmt":"2015-07-29T00:53:45","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1194"},"modified":"2021-10-23T02:56:52","modified_gmt":"2021-10-22T17:56:52","slug":"wsr201401","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201401\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52014\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\u8907\u7d20\u6570 \\(\\alpha = \\dfrac{-1 +\\sqrt{3} i}{2}\\) \u306b\u5bfe\u3057\u3066,\r\n\\[\r\nS _ n = \\textstyle\\sum\\limits _ {k=1}^n {\\alpha}^{k-1} , \\ T _ n = \\textstyle\\sum\\limits _ {k=1}^n k {\\alpha}^{k-1} \\quad ( n = 1, 2, \\cdots )\r\n\\]\r\n\u3068\u304a\u304f. \u305f\u3060\u3057, \\({\\alpha}^0 = 1\\) \u3068\u3059\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(S _ {3m} \\ ( n = 1, 2, \\cdots )\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(T _ {3m} \\ ( n = 1, 2, \\cdots )\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(T _ {2014}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    \\(\\alpha\\) \u306f \\(1\\) \u306e \\(3\\) \u4e57\u6839\u306a\u306e\u3067\r\n\\[\r\n{\\alpha}^3 = 1\r\n\\]\r\n\u307e\u305f, \u3053\u308c\u3092\u5909\u5f62\u3057\u3066\r\n\\[\\begin{align}\r\n( \\alpha -1 ) ( {\\alpha}^2 +\\alpha +1 ) & = 0 \\\\\r\n\\text{\u2234} \\quad {\\alpha}^2 +\\alpha +1 & = 0 \\quad ( \\ \\text{\u2235} \\ \\alpha \\neq 1 \\ )\r\n\\end{align}\\]\r\n

    (1)<\/strong><\/p>\r\n

    \\[\r\nS _ {3m} = m ( {\\alpha}^2 +\\alpha +1 ) = \\underline{0}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\nT _ {3m} & = 1 ( 1 +4 + \\cdots +3m-2 ) \\\\\r\n& \\quad \\alpha ( 2 +5 + \\cdots + 3m-1 ) \\\\\r\n& \\qquad {\\alpha}^2 ( 3 +6 + \\cdots +3m ) \\\\\r\n& = \\dfrac{m ( 3m-1 )}{2} +\\dfrac{m ( 3m+1 ) \\alpha}{2} +\\dfrac{m ( 3m+3 ) {\\alpha}^2}{2} \\\\\r\n& = \\dfrac{m}{2} \\left\\{ 3m-1 +( 3m+1 ) \\dfrac{-1 +\\sqrt{3} i}{2} +( 3m+3 ) \\dfrac{-1 -\\sqrt{3} i}{2} \\right\\} \\\\\r\n& = \\underline{-\\dfrac{m \\left( 3 +\\sqrt{3} i \\right)}{2}}\r\n\\end{align}\\]\r\n

    (3)<\/strong><\/p>\r\n

    \\(2014 = 3 \\cdot 671 +1\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nT _ {2014} & = T _ {3 \\cdot 671} +2014 {\\alpha}^{2013} \\\\\r\n& = -\\dfrac{671 \\left( 3 +\\sqrt{3} i \\right)}{2} +2014 \\\\\r\n& = \\underline{\\dfrac{2015 -671 \\sqrt{3} i}{2}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u8907\u7d20\u6570 \\(\\alpha = \\dfrac{-1 +\\sqrt{3} i}{2}\\) \u306b\u5bfe\u3057\u3066, \\[ S _ n = \\textstyle\\sum\\limits _ {k=1}^n {\\alpha}^{k-1} , \\ … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[120],"tags":[147,112],"class_list":["post-1194","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2014","tag-waseda_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=1194"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1194\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1194"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1194"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}