{"id":406,"date":"2012-05-26T21:01:54","date_gmt":"2012-05-26T12:01:54","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=406"},"modified":"2021-10-23T03:46:22","modified_gmt":"2021-10-22T18:46:22","slug":"wsr201202","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201202\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52012\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u521d\u9805\u3092 \\(a _ 0 \\geqq 0\\) \u3068\u3057, \u4ee5\u4e0b\u306e\u6f38\u5316\u5f0f\u3067\u5b9a\u307e\u308b\u6570\u5217 \\(\\{ a _ n \\} _ {n=0, 1, \\cdots}\\) \u3092\u8003\u3048\u308b.\r\n\\[\r\na _ {n+1} = a _ n -\\left[ \\sqrt{a _ n} \\right] \\quad ( n \\geqq 0 )\r\n\\]\r\n\u305f\u3060\u3057 \\([ x ]\\) \u306f \\(x\\) \u3092\u8d85\u3048\u306a\u3044\u6700\u5927\u306e\u6574\u6570\u3092\u8868\u3059. \u3064\u304e\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a _ 0 =24\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(a _ n = 0\\) \u3068\u306a\u308b\u6700\u5c0f\u306e \\(n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(m\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3057, \\(a _ 0 = m^2\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(1 \\leqq j \\leqq m\\) \u3092\u307f\u305f\u3059 \\(j\\) \u306b\u5bfe\u3057\u3066 \\(a _ {2j-1} , a _ {2j}\\) \u3092 \\(j\\) \u3068 \\(m\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(m\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570, \\(p\\) \u3092 \\(1 \\leqq p \\leqq m-1\\) \u3092\u307f\u305f\u3059\u6574\u6570\u3068\u3057, \\(a _ 0 = m^2-p\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(a _ k = (m-p)^2\\) \u3068\u306a\u308b \\(k\\) \u3092\u6c42\u3081\u3088. \u3055\u3089\u306b, \\(a _ n= 0\\) \u3068\u306a\u308b\u6700\u5c0f\u306e \\(n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \\[\\begin{align}\r\na _ 1 & = 24 -\\left[ \\sqrt{24} \\right] = 24-4 =20 , \\\\\r\na _ 2 & = 20 -\\left[ \\sqrt{20} \\right] = 20-4 =16 , \\\\\r\na _ 3 & = 16 -\\left[ \\sqrt{16} \\right] = 16-4 =12 , \\\\\r\na _ 4 & = 12 -\\left[ \\sqrt{12} \\right] = 12-3 =9 , \\\\\r\na _ 5 & = 9 -\\left[ \\sqrt{9} \\right] = 9-3 =6 , \\\\\r\na _ 6 & = 6 -\\left[ \\sqrt{6} \\right] = 6-2 =4 , \\\\\r\na _ 7 & = 4 -\\left[ \\sqrt{4} \\right] = 4-2 =2 , \\\\\r\na _ 8 & = 2 -\\left[ \\sqrt{2} \\right] = 2-1 =1 , \\\\\r\na _ 9 & = 1 -\\left[ \\sqrt{1} \\right] = 1-1 =0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nn = \\underline{9}\r\n\\]\r\n

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

    \\[\r\na _ {2j-1} = (m-j+1)(m-j) , \\ a _ {2j} =(m-j)^2 \\quad ... [\\text{A}]\r\n\\]\r\n\\(1 \\leqq j \\leqq m\\) \u306b\u3064\u3044\u3066, [A] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(j = 1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\na _ 1 & = m^2 -m = m(m-1) \\\\\r\na _ 2 & = m(m-1) -(m-1) = (m-1)^2\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u3053\u306e\u3068\u304d [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(j = k \\ ( 1 \\leqq k \\leqq m-1 )\\) \u306e\u3068\u304d, [A] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061\r\n\\[\r\na _ {2k-1} = (m-k+1)(m-k) , \\ a _ {2k} = (m-k)^2\r\n\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\na _ {2k+1} & = (m-k)^2 -(m-k) \\\\\r\n& = (m-k)(m-k-1) , \\\\\r\na _ {2k+2} & =(m-k)(m-k-1) -(m-k-1) \\\\\r\n& = (m-k-1)^2\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(j = k+1\\) \u306e\u3068\u304d\u3082 [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      \u3088\u3063\u3066\r\n\\[\r\na _ {2j-1} = \\underline{(m-j)(m-j+1)} , \\ a _ {2j} = \\underline{(m-j)}^2\r\n\\]\r\n

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

      \\[\r\na _ {2j} = (m-j)^2-p+j \\quad ... [\\text{B}]\r\n\\]\r\n\\(0 \\leqq j \\leqq p\\) \u306b\u3064\u3044\u3066, [B] \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(j = 0\\) \u306e\u3068\u304d
        \r\n\\(a _ 0=m^2-p\\) \u306a\u306e\u3067, [B] \u306f\u6210\u7acb\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n

      2. 2*<\/strong>\u3000\\(j = l \\ ( 0 \\leqq k \\leqq p-1 )\\) \u306e\u3068\u304d, [B] \u304c\u6210\u7acb\u3059\u308b, \u3059\u306a\u308f\u3061\r\n\\[\r\na _ {2l} = (m-l)^2-p+l\r\n\\]\r\n\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\na _ {2l+1} & = (m-l)^2-p+l -(m-l-1) \\\\\r\n& = (m-l)(m-l-1) -p+l+1 , \\\\\r\na _ {2l+2} & = (m-l)(m-l-1) -p+l+1 -(m-l-1) \\\\\r\n& = (m-l-1)^2 -p+l+1\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(j = l+1\\) \u306e\u3068\u304d\u3082 [B] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a, \\(0 \\leqq j \\leqq p\\) \u306b\u3064\u3044\u3066\r\n\\[\r\na _ {2j} = (m-j)^2-p+j\r\n\\]\r\n\u3088\u3063\u3066, \\(a _ {2p} = (m-p)^2\\) \u306a\u306e\u3067\r\n\\[\r\nk = \\underline{2p}\r\n\\]\r\n(2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(a _ 0 =m^2\\) \u306b\u5bfe\u3057\u3066, \\(a _ i = 0\\) \u3068\u306a\u308b\u6700\u5c0f\u306e \\(i\\) \u306f\r\n\\[\r\ni = 2m-1\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5c0f\u306e \\(n\\) \u306e\u5024\u306f\r\n\\[\r\n2p +2(m-p)-1 = \\underline{2m-1}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u521d\u9805\u3092 \\(a _ 0 \\geqq 0\\) \u3068\u3057, \u4ee5\u4e0b\u306e\u6f38\u5316\u5f0f\u3067\u5b9a\u307e\u308b\u6570\u5217 \\(\\{ a _ n \\} _ {n=0, 1, \\cdots}\\) \u3092\u8003\u3048\u308b. \\[ a _ {n+1} = a _ n -\\left[ \\ … \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":[64],"tags":[147,68],"class_list":["post-406","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2012","tag-waseda_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/406","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=406"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/406\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=406"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=406"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=406"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}