{"id":784,"date":"2013-05-08T22:44:12","date_gmt":"2013-05-08T13:44:12","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=784"},"modified":"2021-03-23T17:14:52","modified_gmt":"2021-03-23T08:14:52","slug":"kyr201303","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr201303\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb2013\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3057, \u6574\u5f0f \\(x^n\\) \u3092\u6574\u5f0f \\(x^2-2x-1\\) \u3067\u5272\u3063\u305f\u4f59\u308a\u3092 \\(ax+b\\) \u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d \\(a\\) \u3068 \\(b\\) \u306f\u6574\u6570\u3067\u3042\u308a, \u3055\u3089\u306b\u305d\u308c\u3089\u3092\u3068\u3082\u306b\u5272\u308a\u5207\u308b\u7d20\u6570\u306f\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u3092\u793a\u305b.<\/p>\r\n

\r\n\r\n

\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

\\(x _ n\\) \u3092\u6574\u5f0f \\(x^2-2x-1\\) \u3067\u5272\u3063\u305f\u5546\u3092 \\(P(x\\) ) , \u4f59\u308a\u3092 \\(a _ n x+b _ n\\) \u3068\u304a\u304f.
\r\n\\(n=1\\) \u306e\u3068\u304d\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\r\n\\[\r\na _ 1 = 1 , \\ b _ 1 = 0 \\quad ... [1]\r\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nx^{n+1} & = x \\left\\{ (x^2-2x-1) P(x) +a _ n x +b _ n \\right\\} \\\\\r\n& = (x^2-2x-1) \\left\\{ x P(x) +a _ n \\right\\} +( 2 a _ n +b _ n )x +a _ n\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\na _ {n+1} = 2 a _ n +b _ n , \\ b _ {n+1} = a _ n \\quad ... [2]\r\n\\]\r\n\u307e\u305a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066<\/p>\r\n

    \r\n
  1. [A] : \u300e \\(a _ n , b _ n\\) \u306f\u6574\u6570\u3067\u3042\u308b. \u300f<\/li>\r\n<\/ol>\r\n

    \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\\(n=1\\) \u306e\u3068\u304d
      \r\n[1] \u3088\u308a, [A] \u304c\u6210\u7acb\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(n=k\\) \u306e\u3068\u304d, [A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068
      \r\n[2] \u3088\u308a, \\(a _ {k+1} , b _ {k+1}\\) \u3082\u6574\u6570\u3068\u306a\u308a, \\(n = k+1\\) \u306e\u3068\u304d\u3082, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, [A]\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u3055\u308c\u305f.
      \r\n\u7d9a\u3044\u3066, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066<\/p>\r\n

        \r\n
      1. [B] : \u300e \\(a _ n\\) \u3068 \\(b _ n\\) \u3092\u3068\u3082\u306b\u5272\u5207\u308b\u7d20\u6570\u306f\u306a\u3044. \u300f<\/li>\r\n<\/ol>\r\n

        \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u3059.<\/p>\r\n

          \r\n

        1. 1*<\/strong>\u3000\\(n=1\\) \u306e\u3068\u304d
          \r\n[1] \u3088\u308a [B] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

        2. 2*<\/strong>\u3000\\(n \\geqq 2\\) \u306e\u3068\u304d\u306b\u3064\u3044\u3066, \u80cc\u7406\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.
          \r\n\\(a _ n = kp\\) , \\(b _ n = mp\\) \uff08 \\(p\\) \u306f\u7d20\u6570, \\(n , m\\) \u306f\u6574\u6570\uff09\u3068\u8868\u305b\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068, [2] \u3088\u308a\r\n\\[\\begin{align}\r\na _ {n-1} & = b _ n = mp , \\\\\r\nb _ {n-1} & = a _ n -2 b _ n = (k+2m)p\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(a _ {n-1} , b _ {n-1}\\) \u3082\u7d20\u6570 \\(p\\) \u3067\u5272\u5207\u308c\u308b.
          \r\n\u3057\u304b\u3057, \u3053\u308c\u3092\u7e70\u8fd4\u3059\u3068, \\(a _ {1} , b _ {1}\\) \u3082\u7d20\u6570 \\(p\\) \u3067\u5272\u5207\u308c\u308b\u3053\u3068\u306b\u306a\u308b\u304c, \u3053\u308c\u306f\u77db\u76fe\u3067\u3042\u308b.
          \r\n\u3086\u3048\u306b, [B] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

          \u4ee5\u4e0a\u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, [B] \u304c\u6210\u7acb\u3059\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3057, \u6574\u5f0f \\(x^n\\) \u3092\u6574\u5f0f \\(x^2-2x-1\\) \u3067\u5272\u3063\u305f\u4f59\u308a\u3092 \\(ax+b\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d \\(a\\) \u3068 \\(b\\) \u306f\u6574\u6570\u3067\u3042\u308a, \u3055\u3089\u306b\u305d\u308c\u3089\u3092\u3068\u3082\u306b\u5272\u308a\u5207\u308b\u7d20\u6570\u306f\u5b58\u5728 … \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":[88],"tags":[140,111],"class_list":["post-784","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2013","tag-kyoto_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/784","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=784"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/784\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=784"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=784"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=784"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}