{"id":3078,"date":"2017-03-13T06:00:08","date_gmt":"2017-03-13T04:00:08","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=3078"},"modified":"2022-03-26T11:31:41","modified_gmt":"2022-03-26T09:31:41","slug":"numeros-de-catalan","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-de-catalan\/","title":{"rendered":"N\u00fameros de Catalan"},"content":{"rendered":"<p>Los <a href=\"http:\/\/bit.ly\/2lRbVUL\">n\u00fameros de Catalan<\/a> forman la sucesi\u00f3n cuyo t\u00e9rmino general es<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_1.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_1.png?resize=559%2C98\" alt=\"Numeros_de_Catalan_1\" width=\"559\" height=\"98\" class=\"aligncenter size-full wp-image-3079\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_1.png?w=559&amp;ssl=1 559w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_1.png?resize=300%2C52&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_1.png?resize=100%2C17&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_1.png?resize=150%2C26&amp;ssl=1 150w\" sizes=\"(max-width: 559px) 100vw, 559px\" data-recalc-dims=\"1\" \/><\/a><\/p>\n<p>Los primeros n\u00fameros de Catalan son<\/p>\n<pre lang=\"text\">\n   1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786\n<\/pre>\n<p>Los n\u00fameros de Catalan satisfacen la siguiente relaci\u00f3n de recurrencia:<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_2.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_2.png?resize=393%2C84\" alt=\"Numeros_de_Catalan_2\" width=\"393\" height=\"84\" class=\"aligncenter size-full wp-image-3080\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_2.png?w=393&amp;ssl=1 393w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_2.png?resize=300%2C64&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_2.png?resize=100%2C21&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_2.png?resize=150%2C32&amp;ssl=1 150w\" sizes=\"(max-width: 393px) 100vw, 393px\" data-recalc-dims=\"1\" \/><\/a><\/p>\n<p>Asint\u00f3ticamente, los n\u00fameros de Catalan crecen como:<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png?resize=194%2C89\" alt=\"Numeros_de_Catalan_3\" width=\"194\" height=\"89\" class=\"aligncenter size-full wp-image-3081\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png?w=194&amp;ssl=1 194w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png?resize=100%2C45&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png?resize=150%2C68&amp;ssl=1 150w\" sizes=\"(max-width: 194px) 100vw, 194px\" data-recalc-dims=\"1\" \/><\/a><br \/>\nconsiderando que el cociente entre el n-\u00e9simo n\u00famero de Catalan y la expresi\u00f3n de la derecha tiende hacia 1 cuando n tiende a infinito.<\/p>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   catalan :: [Integer]\n   grafica :: Int -> Int -> IO ()\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>catalan es la lista de t\u00e9rminos de la sucesi\u00f3n de Catalan. Por ejemplo, <\/li>\n<\/ul>\n<pre lang=\"text\">\n     \u03bb> take 12 catalan\n     [1,1,2,5,14,42,132,429,1430,4862,16796,58786]\n     \u03bb> length (show (catalan !! 50000))\n     30096\n<\/pre>\n<ul>\n<li>(grafica a b) dibuja los n-\u00e9simos t\u00e9rminos de la sucesi\u00f3n de Catalan, para n entre a y b, junto con los de la expresi\u00f3n de la derecha de<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png?resize=194%2C89\" alt=\"Numeros_de_Catalan_3\" width=\"194\" height=\"89\" class=\"aligncenter size-full wp-image-3081\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png?w=194&amp;ssl=1 194w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png?resize=100%2C45&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_3.png?resize=150%2C68&amp;ssl=1 150w\" sizes=\"(max-width: 194px) 100vw, 194px\" data-recalc-dims=\"1\" \/><\/a><br \/>\nPor ejemplo, (grafica 5 10) dibuja<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_4.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_4.png?resize=640%2C480\" alt=\"Numeros_de_Catalan_4\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-3082\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_4.png?w=640&amp;ssl=1 640w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_4.png?resize=300%2C225&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_4.png?resize=100%2C75&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_4.png?resize=150%2C112&amp;ssl=1 150w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/a><br \/>\ny (grafica 55 60) dibuja<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_5.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_5.png?resize=640%2C480\" alt=\"Numeros_de_Catalan_5\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-3083\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_5.png?w=640&amp;ssl=1 640w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_5.png?resize=300%2C225&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_5.png?resize=100%2C75&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2017\/03\/Numeros_de_Catalan_5.png?resize=150%2C112&amp;ssl=1 150w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/a><\/li>\n<\/ul>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Graphics.Gnuplot.Simple\n\ncatalan :: [Integer]\ncatalan = scanl (\\c n -> c*2*(2*n-1) `div` (n+1)) 1 [1..]\n\ngrafica :: Int -> Int -> IO ()\ngrafica a b = \n  plotLists [Key Nothing]\n            [[(fromIntegral n, fromIntegral (catalan !! n)) | n <- [a..b]]\n            ,[(n,4**n\/(n**(3\/2)*(sqrt pi))) | n <- [c..d]]]\n  where c, d :: Double\n        c = fromIntegral a\n        d = fromIntegral b\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Los n\u00fameros de Catalan forman la sucesi\u00f3n cuyo t\u00e9rmino general es Los primeros n\u00fameros de Catalan son 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786 Los n\u00fameros de Catalan satisfacen la siguiente relaci\u00f3n de recurrencia: Asint\u00f3ticamente, los n\u00fameros de Catalan crecen como: considerando que el cociente entre el n-\u00e9simo n\u00famero de&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[4],"tags":[30,183,376,11,366,309,78,236],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3078"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=3078"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3078\/revisions"}],"predecessor-version":[{"id":3111,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3078\/revisions\/3111"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3078"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3078"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3078"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}