{"id":5167,"date":"2019-11-25T05:30:16","date_gmt":"2019-11-25T03:30:16","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5167"},"modified":"2022-03-25T20:06:34","modified_gmt":"2022-03-25T18:06:34","slug":"numeros-triangulares","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-triangulares\/","title":{"rendered":"N\u00fameros triangulares"},"content":{"rendered":"<p>La sucesi\u00f3n de los <a href=\"http:\/\/bit.ly\/16xJtKZ\">n\u00fameros triangulares<\/a> se obtiene sumando los n\u00fameros naturales.<\/p>\n<pre lang=\"text\">\n   *     *      *        *         *   \n        * *    * *      * *       * *  \n              * * *    * * *     * * * \n                      * * * *   * * * *\n                               * * * * * \n   1     3      6        10        15\n<\/pre>\n<p>As\u00ed, los 5 primeros n\u00fameros triangulares son<\/p>\n<pre lang=\"text\">\n    1 = 1\n    3 = 1+2\n    6 = 1+2+3\n   10 = 1+2+3+4\n   15 = 1+2+3+4+5\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   triangulares :: [Integer]\n<\/pre>\n<p>tal que triangulares es la lista de los n\u00fameros triangulares. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   take 10 triangulares  ==  [1,3,6,10,15,21,28,36,45,55]\n   maximum (take (5*10^6) triangulares4)  ==  12500002500000\n<\/pre>\n<p>Comprobar con QuickCheck que entre dos n\u00fameros triangulares consecutivos siempre hay un n\u00famero primo.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck (Property, (==>), quickCheck)\nimport Data.Numbers.Primes (primes)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\ntriangulares :: [Integer]\ntriangulares = [sum [1..n] | n <- [1..]]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\ntriangulares2 :: [Integer]\ntriangulares2 = map triangular [1..]\n\n-- (triangular n) es el n-\u00e9simo n\u00famero triangular. Por ejemplo, \n--    triangular 5  ==  15\ntriangular :: Integer -> Integer\ntriangular 1 = 1\ntriangular n = n + triangular (n-1)\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\ntriangulares3 :: [Integer]\ntriangulares3 = 1 : [x+y | (x,y) <- zip [2..] triangulares]\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\ntriangulares4 :: [Integer]\ntriangulares4 = scanl1 (+) [1..]\n\n-- 5\u00aa soluci\u00f3n\n-- ===========\n\ntriangulares5 :: [Integer]\ntriangulares5 = [(n*(n+1)) `div` 2 | n <- [1..]]\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> maximum (take (10^4) triangulares)\n--    50005000\n--    (2.10 secs, 8,057,774,104 bytes)\n--    \u03bb> maximum (take (10^4) triangulares2)\n--    50005000\n--    (18.89 secs, 12,142,690,784 bytes)\n--    \u03bb> maximum (take (10^4) triangulares3)\n--    50005000\n--    (0.01 secs, 4,600,976 bytes)\n--    \u03bb> maximum (take (10^4) triangulares4)\n--    50005000\n--    (0.01 secs, 3,643,192 bytes)\n--    \u03bb> maximum (take (10^4) triangulares5)\n--    50005000\n--    (0.02 secs, 5,161,464 bytes)\n--    \n--    \u03bb> maximum (take (3*10^4) triangulares3)\n--    450015000\n--    (26.06 secs, 72,546,027,136 bytes)\n--    \u03bb> maximum (take (3*10^4) triangulares4)\n--    450015000\n--    (0.02 secs, 10,711,600 bytes)\n--    \u03bb> maximum (take (3*10^4) triangulares5)\n--    450015000\n--    (0.03 secs, 15,272,320 bytes)\n--    \n--    \u03bb> maximum (take (5*10^6) triangulares4)\n--    12500002500000\n--    (1.67 secs, 1,772,410,336 bytes)\n--    \u03bb> maximum (take (5*10^6) triangulares5)\n--    12500002500000\n--    (4.09 secs, 2,532,407,720 bytes)\n\n-- La propiedad es\nprop_triangulares :: Int -> Property\nprop_triangulares n =\n  n >= 0 ==> siguientePrimo x < y\n  where (x:y:_) = drop n triangulares4\n\n-- (siguientePrimo n) es el menor primo mayor o igual que n. Por\n-- ejemplo, \n--    siguientePrimo 14  ==  17\n--    siguientePrimo 17  ==  17\nsiguientePrimo :: Integer -> Integer\nsiguientePrimo n = head (dropWhile (< n) primes)\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_triangulares\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nAutores, la escena acaba<br \/>\ncon un dogma de teatro:<br \/>\nEn el principio era la m\u00e1scara.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>La sucesi\u00f3n de los n\u00fameros triangulares se obtiene sumando los n\u00fameros naturales. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 1 3 6 10 15 As\u00ed, los 5&#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":[5],"tags":[8,30,415,10,11,6,252,40,9],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5167"}],"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=5167"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5167\/revisions"}],"predecessor-version":[{"id":5218,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5167\/revisions\/5218"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5167"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5167"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}