{"id":1189,"date":"2015-07-25T21:35:09","date_gmt":"2015-07-25T12:35:09","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1189"},"modified":"2021-10-20T13:57:26","modified_gmt":"2021-10-20T04:57:26","slug":"ykr201404","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201404\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2014\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\u5e73\u9762\u4e0a\u306b\u534a\u5f84 \\(1\\) \u3068\u534a\u5f84 \\(2\\) \u306e\u540c\u5fc3\u5186 \\(C _ 1\\) \u3068 \\(C _ 2\\) \u304c\u3042\u308b.\r\n\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(C _ 2\\) \u306e\u5468\u3092 \\(3n\\) \u7b49\u5206\u3059\u308b \\(3n\\) \u500b\u306e\u70b9\u304c\u3042\u308b. \u3053\u306e \\(3n\\) \u500b\u306e\u70b9\u306e\u4e2d\u304b\u3089\u7570\u306a\u308b \\(3\\) \u70b9\u3092\u9078\u3076\u3068\u304d, \u6b21\u306e\uff08\uff0a\uff09\u3092\u307f\u305f\u3059\u9078\u3073\u65b9\u306e\u7dcf\u6570\u3092 \\(a _ k \\ ( k = 0, 1, 2, 3 )\\) \u3068\u3059\u308b.<\/p>\r\n<ol>\r\n<li>\uff08\uff0a\uff09\u9078\u3093\u3060 \\(3\\) \u70b9\u3092\u9802\u70b9\u3068\u3059\u308b\u4e09\u89d2\u5f62\u306e\u8fba\u306e\u3046\u3061, \u3061\u3087\u3046\u3069 \\(k\\) \u500b\u304c \\(C _ 1\\) \u306e\u5468\u3068\u5171\u6709\u70b9\u3092\u3082\u3064.<\/li>\r\n<\/ol>\r\n<p>\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(n = 2\\) \u306e\u3068\u304d, \\(a _ 0 , a _ 1 , a _ 2 , a _ 3\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(n \\geqq 2\\) \u306e\u3068\u304d, \\(a _ 0 , a _ 1 , a _ 2 , a _ 3\\) \u3092 \\(n\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\(6\\) \u3064\u306e\u70b9\u304b\u3089 \\(3\\) \u70b9\u3092\u9078\u3076\u306e\u3067, \u4e09\u89d2\u5f62\u306f\u5168\u90e8\u3067 \\({} _ {6} \\text{C} {} _ 3 = 20\\) \u901a\u308a\u3042\u308b.<br \/>\r\n\\(6\\) \u3064\u306e\u70b9\u306b \\(0 , 1, \\cdots , 5\\) \u3068\u756a\u53f7\u3092\u3064\u3051\u308b.<br \/>\r\n\u307e\u305a\u70b9 \\(0\\) \u3092\u5fc5\u305a\u9078\u3076\u3068\u4eee\u5b9a\u3057\u3066, \u4e92\u3044\u306b\u5408\u540c\u306a\u4e09\u89d2\u5f62\u306b\u306a\u3089\u306a\u3044\u3088\u3046\u306a\u6b8b\u308a \\(2\\) \u70b9\u306e\u9078\u3073\u65b9\u3092\u8003\u3048\u305f\u4e0a\u3067, \u3053\u306e\u5834\u5408\u306e\u6570\u3092 \\(6\\) \u500d\u3059\u308c\u3070\u3088\u3044. \uff08\u305f\u3060\u3057, \u3053\u306e\u4e2d\u306b\u540c\u3058\u70b9\u306e\u9078\u3073\u65b9\u3092\u3059\u308b\u3082\u306e\u304c\u306a\u3044\u304b\u6ce8\u610f\u3059\u308b. \uff09<br \/>\r\n\u4e09\u89d2\u5f62\u306e\u8fba\u3068\u306a\u308a\u3046\u308b\u7dda\u5206\u304c\u4e0b\u56f3\u306e\u70b9\u7dda\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066, \u5404\u5834\u5408\u306b\u3064\u3044\u3066\u8003\u3048\u308b.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/ykr20140401.svg\" alt=\"ykr20140401\" class=\"aligncenter size-full\" \/>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(a _ 0\\) \u306b\u3064\u3044\u3066<br \/>\r\n\u3069\u306e\u3088\u3046\u306b \\(3\\) \u70b9\u3092\u9078\u3093\u3067\u3082, \u5c11\u306a\u304f\u3068\u3082 \\(1\\) \u8fba\u306f \\(C _ 1\\) \u3068\u5171\u6709\u70b9\u3092\u3082\u3064\u306e\u3067\r\n\\[\r\na _ 0 = 0\r\n\\]<\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(a _ 1\\) \u306b\u3064\u3044\u3066<br \/>\r\n\u70b9 \\(( 0, 1, 2 )\\) \u3068\u9078\u3093\u3060\u5834\u5408\u306e\u307f\u304c\u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u3067\r\n\\[\r\na _ 1 = 6 \\cdot 1 = 6\r\n\\]<\/li>\r\n<li><p><strong>3*<\/strong>\u3000\\(a _ 3\\) \u306b\u3064\u3044\u3066<br \/>\r\n\u70b9 \\(( 0, 2, 4 )\\) \u3068\u9078\u3093\u3060\u5834\u5408\u306e\u307f\u304c\u6761\u4ef6\u3092\u307f\u305f\u3059.<br \/>\r\n\u3053\u308c\u306f\u6b63\u4e09\u89d2\u5f62\u306a\u306e\u3067, \u56de\u8ee2\u3059\u308b\u3068 \\(3\\) \u3064\u304c\u91cd\u8907\u3059\u308b\u306e\u3067\r\n\\[\r\na _ 3 = \\dfrac{6 \\cdot 1}{3} = 2\r\n\\]<\/li>\r\n<li><p><strong>4*<\/strong>\u3000\\(a _ 2\\) \u306b\u3064\u3044\u3066<br \/>\r\n<strong>1*<\/strong>\uff5e<strong>3*<\/strong>\u4ee5\u5916\u306e\u5834\u5408\u3092\u8003\u3048\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\r\na _ 2 = 20 -( 0 +6 +2 ) = 12\r\n\\]<\/li>\r\n<\/ol>\r\n<p>\u4ee5\u4e0a\u3088\u308a\r\n\\[\r\na _ 0 = \\underline{0} , \\ a _ 1 = \\underline{6} , \\ a _ 2 = \\underline{12} , \\ a _ 3 = \\underline{2}\r\n\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\\(3n\\) \u3064\u306e\u70b9\u304b\u3089 \\(3\\) \u70b9\u3092\u9078\u3076\u306e\u3067, \u4e09\u89d2\u5f62\u306f\u5168\u90e8\u3067 \\({} _ {3n} \\text{C} {} _ 3 = \\dfrac{1}{2} n (3n-1) (3n-2)\\) \u901a\u308a\u3042\u308b.<br \/>\r\n\u70b9 \\(0 , k \\ ( 1 \\leqq k \\leqq 3n-1 )\\) \u3092\u7d50\u3076\u8fba\u304c \\(C _ 1\\) \u3068\u5171\u6709\u70b9\u3092\u3082\u3064\u6761\u4ef6\u306f\r\n\\[\r\nn \\leqq k-1 \\leqq 2n\r\n\\]\r\n\u3053\u308c\u306b\u6ce8\u610f\u3057\u3066, <strong>(1)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308b.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(a _ 0\\) \u306b\u3064\u3044\u3066<br \/>\r\n\\(2\\) \u70b9 \\(b , c\\) \u3092\r\n\\[\r\n1 \\leqq c \\lt b \\leqq n-1\r\n\\]\r\n\u3068\u306a\u308b\u3088\u3046\u306b\u9078\u3079\u3070\u3088\u3044\u306e\u3067\r\n\\[\r\na _ 0 = 3n {} _ {n-1} \\text{C} {} _ 2 = \\dfrac{3}{2} n (n-1) (n-2)\r\n\\]<\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(a _ 1\\) \u306b\u3064\u3044\u3066<br \/>\r\n\\(2\\) \u70b9 \\(b , c\\) \u3092\r\n\\[\r\nn \\leqq b \\leqq 2n-1 , \\ b-n+1 \\leqq c \\leqq n-1\r\n\\]\r\n\u3068\u306a\u308b\u3088\u3046\u306b\u9078\u3079\u3070\u3088\u3044\u306e\u3067\r\n\\[\\begin{align}\r\na _ 1 & = 3n \\textstyle\\sum\\limits _ {b=n}^{2n-1} ( 2n -b -1 ) \\\\\r\n& = 3n \\left\\{ (n-1) +(n-2) + \\cdots +1 +0 \\right\\} \\\\\r\n& = 3n \\cdot \\dfrac{n (n-1)}{2} \\\\\r\n& = \\dfrac{3}{2} n^2 (n-1)\r\n\\end{align}\\]<\/li>\r\n<li><p><strong>3*<\/strong>\u3000\\(a _ 3\\) \u306b\u3064\u3044\u3066<br \/>\r\n\u70b9 \\(( 0, n, 2n )\\) \u3068\u9078\u3093\u3060\u5834\u5408\u306e\u307f\u304c\u6761\u4ef6\u3092\u307f\u305f\u3059.<br \/>\r\n\u3053\u308c\u306f\u6b63\u4e09\u89d2\u5f62\u306a\u306e\u3067, \u56de\u8ee2\u3059\u308b\u3068 \\(3\\) \u3064\u304c\u91cd\u8907\u3059\u308b\u306e\u3067\r\n\\[\r\na _ 3 = \\dfrac{3n \\cdot 1}{3} = n\r\n\\]<\/li>\r\n<li><p><strong>4*<\/strong>\u3000\\(a _ 2\\) \u306b\u3064\u3044\u3066<br \/>\r\n<strong>1*<\/strong>\uff5e<strong>3*<\/strong>\u4ee5\u5916\u306e\u5834\u5408\u3092\u8003\u3048\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\\begin{align}\r\na _ 2 & = \\dfrac{n}{2} \\left\\{ (3n-1) (3n-2) -3 (n-1) (n-2) -3n (n-1) -2 \\right\\} \\\\\r\n& = \\dfrac{n}{2} ( 3n^2 -3n -6 ) \\\\\r\n& = \\dfrac{3}{2} n (n-2) (n+1)\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n<p>\u4ee5\u4e0a\u3088\u308a\r\n\\[\\begin{align}\r\na _ 0 & = \\underline{\\dfrac{3}{2} n (n-1) (n-2)} , \\quad a _ 1 = \\underline{\\dfrac{3}{2} n^2 (n-1)} , \\\\\r\n& \\qquad a _ 2 = \\underline{\\dfrac{3}{2} n (n-2) (n+1)} , \\quad a _ 3 = \\underline{n}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5e73\u9762\u4e0a\u306b\u534a\u5f84 \\(1\\) \u3068\u534a\u5f84 \\(2\\) \u306e\u540c\u5fc3\u5186 \\(C _ 1\\) \u3068 \\(C _ 2\\) \u304c\u3042\u308b. \u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, \\(C _ 2\\) \u306e\u5468\u3092 \\(3n\\) \u7b49\u5206\u3059\u308b \\(3n\\) \u500b\u306e\u70b9\u304c\u3042\u308b &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/ykr201404\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[122],"tags":[112,9],"class_list":["post-1189","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2014","tag-112","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1189","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=1189"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1189\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1189"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1189"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}