{"id":1996,"date":"2021-11-23T09:25:11","date_gmt":"2021-11-23T00:25:11","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1996"},"modified":"2021-11-23T09:56:05","modified_gmt":"2021-11-23T00:56:05","slug":"ngr202104","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr202104\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2021\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(0 \\leqq a \\lt 1\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570 \\(a\\) \u306b\u5bfe\u3057, \u6570\u5217 \\(\\{ a_n \\}\\) \u3092\r\n\\[\r\na_1 = a , \\qquad a _ {n+1} = 3 \\left[ a_n +\\dfrac{1}{2} \\right] -2 a_n \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u3068\u3044\u3046\u6f38\u5316\u5f0f\u3067\u5b9a\u3081\u308b.\r\n\u305f\u3060\u3057 \\([x]\\) \u306f \\(x\\) \u4ee5\u4e0b\u306e\u6700\u5927\u306e\u6574\u6570\u3092\u8868\u3059.\r\n\u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a\\) \u304c \\(0 \\leqq a \\lt 1\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \u70b9 \\(( x , y ) = ( a_1 , a_2 )\\) \u306e\u8ecc\u8de1\u3092 \\(xy\\) \u5e73\u9762\u4e0a\u306b\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(a_n -[ a_n ] \\geqq \\dfrac{1}{2}\\) \u306a\u3089\u3070, \\(a_n \\lt a _ {n+1}\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a_n \\gt a _ {n+1}\\) \u306a\u3089\u3070,\r\n\\(a _ {n+1} = 3 [ a_n ] -2 a_n\\) \u304b\u3064 \\([ a _ {n+1} ] = [ a_n ] -1\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  4. (4)<\/strong>\u3000\u3042\u308b \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570 \\(k\\) \u306b\u5bfe\u3057\u3066, \\(a_1 \\gt a_2 \\gt \\cdots \\gt a_k\\) \u304c\u6210\u308a\u7acb\u3064\u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d \\(a_k\\) \u3092 \\(a\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\ny & = 3 \\left[ x +\\dfrac{1}{2} \\right] -2x \\\\\r\n& = \\left\\{ \\begin{array}{ll} -2x & \\left( \\ 0 \\leqq a \\lt \\dfrac{1}{2} \\text{\u306e\u3068\u304d} \\ \\right) \\\\ -2x +3 & \\left( \\ \\dfrac{1}{2} \\leqq a \\lt 1 \\text{\u306e\u3068\u304d} \\ \\right) \\end{array} \\right.\r\n\\end{align}\\]\r\n

    \u3088\u3063\u3066, \u6c42\u3081\u308b\u8ecc\u8de1\u306f\u4e0b\u56f3.<\/p>\r\n\"ngr20210401\"\r\n

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

    \u4e00\u822c\u306b \\([ a_n ] \\leqq a_n \\lt [ a_n ] +1\\) \u306a\u306e\u3067, \u6761\u4ef6\u3068\u3042\u308f\u305b\u3066\r\n\\[\r\n[ a_n ] +\\dfrac{1}{2} \\leqq a_n \\lt [ a_n ] +1\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\na _ {n+1} & = 3 \\left[ a_n +\\dfrac{1}{2} \\right] -2 a_n \\\\\r\n& \\geqq 3 \\left[ [ a_n ] +1 \\right] -2 a_n \\\\\r\n& = 3 \\left( [ a_n ] +1 \\right) -2 a_n \\\\\r\n& \\gt 3 a_n -2 a_n = a_n\r\n\\end{align}\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\na_n \\lt a _ {n+1}\r\n\\]\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u306e\u5bfe\u5076\u3092\u3068\u308c\u3070\r\n\\[\r\na _ n \\geqq a _ {n+1} \\ \\Rightarrow \\ a_n -[ a_n ] \\lt \\dfrac{1}{2}\r\n\\]\r\n\u3086\u3048\u306b, \u6761\u4ef6\u3088\u308a\r\n\\[\r\n[ a_n ] \\leqq a_n \\lt [ a_n ] +\\dfrac{1}{2}\r\n\\]\r\n\u3053\u306e\u3046\u3061, \\(a_ n = [ a_n ]\\) \u306e\u3068\u304d, \\(a_n\\) \u306f\u6574\u6570\u3067 \\(\\left[ a_n +\\dfrac{1}{2} \\right] = a_n\\) \u3068\u306a\u308a\r\n\\[\r\na _ {n+1} = 3 a_n -2 a_n = a_n\r\n\\]\r\n\u3053\u308c\u306f, \u6761\u4ef6\u3092\u307f\u305f\u3055\u305a\u4e0d\u9069.
    \r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n[ a_n ] & \\lt a_n \\lt [ a_n ] +\\dfrac{1}{2} \\quad ... [1] \\\\\r\n\\text{\u2234} \\quad [ a_n ] +\\dfrac{1}{2} & \\lt a_n +\\dfrac{1}{2} \\lt [ a_n ] +1\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\left[ a_n +\\dfrac{1}{2} \\right] = [ a_n ]\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\na _ {n+1} = \\underline{3 [ a_n ] -2 a_n} \\quad ... [2]\r\n\\]\r\n[1] \u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\na _ {n+1} & \\gt 3 [ a_n ] -2 \\left( [ a_n ] +\\dfrac{1}{2} \\right) = [ a_n ] -1 \\ , \\\\\r\na _ {n+1} & \\lt 3 [ a_n ] -2 [ a_n ] = [ a_n ]\r\n\\end{align}\\]\r\n\u3059\u306a\u308f\u3061\r\n\\[\r\n[ a_n ] -1 \\lt a _ {n+1} \\lt [ a_n ]\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n[ a _ {n+1} ] = \\underline{[ a_n ] -1} \\quad ... [3]\r\n\\]\r\n

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

    \\(n = 1 , 2 , \\cdots , k-1\\) \u306b\u3064\u3044\u3066, \\(a_n \\lt a _ {n+1}\\) \u306a\u306e\u3067, [3] \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n[ a _ {n+1} ] = [ a_n ] -1 \\quad ( n = 1 , 2 , \\cdots , k-1 )\r\n\\]\r\n\u6570\u5217 \\(\\{ [ a_n ] \\}\\) \u306f, \u521d\u9805 \\([ a_1 ] = [ a ] = 0\\) , \u516c\u5dee \\(-1\\) \u306e\u7b49\u5dee\u6570\u5217\u306a\u306e\u3067\r\n\\[\r\n[ a_n ] = -n+1\r\n\\]\r\n[2] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\na _ {n+1} & = -3n +3 -2 a_n \\\\\r\n\\text{\u2234} \\quad a _ {n+1} +(n+1) -\\dfrac{4}{3} & = -2 \\left( a_n +n -\\dfrac{4}{3} \\right)\r\n\\end{align}\\]\r\n\u6570\u5217 \\(\\left\\{ a_n +n -\\dfrac{4}{3} \\right\\}\\) \u306f, \u521d\u9805 \\(a_1 +1 -\\dfrac{4}{3} = a -\\dfrac{1}{3}\\) , \u516c\u6bd4 \\(-2\\) \u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\r\n\\[\\begin{align}\r\na_n +n -\\dfrac{4}{3} & = \\left( a -\\dfrac{1}{3} \\right) (-2)^{n-1} \\\\\r\n\\text{\u2234} \\quad a_n & = \\left( a -\\dfrac{1}{3} \\right) (-2)^{n-1} -n +\\dfrac{4}{3}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\na_k = \\underline{\\left( a -\\dfrac{1}{3} \\right) (-2)^{k-1} -k +\\dfrac{4}{3}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(0 \\leqq a \\lt 1\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570 \\(a\\) \u306b\u5bfe\u3057, \u6570\u5217 \\(\\{ a_n \\}\\) \u3092 \\[ a_1 = a , \\qquad a _ {n+1} = 3 \\left[ a_n +\\dfrac{ … \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":[176],"tags":[143,165],"class_list":["post-1996","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2021","tag-nagoya_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1996","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=1996"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1996\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1996"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1996"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1996"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}