{"id":5904,"date":"2020-05-19T07:43:37","date_gmt":"2020-05-19T05:43:37","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5904"},"modified":"2020-05-26T07:46:08","modified_gmt":"2020-05-26T05:46:08","slug":"numeros-de-perrin","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-de-perrin\/","title":{"rendered":"N\u00fameros de Perrin"},"content":{"rendered":"<p>Los <a href=\"https:\/\/en.wikipedia.org\/wiki\/Perrin_number\">n\u00fameros de Perrin<\/a> se definen por la elaci\u00f3n de recurrencia<\/p>\n<pre lang=\"text\"> \n   P(n) = P(n - 2) + P(n - 3) si n > 2,\n<\/pre>\n<p>con los valores iniciales<\/p>\n<pre lang=\"text\"> \n   P(0) = 3, P(1) = 0 y P(2) = 2.\n<\/pre>\n<p>Definir la sucesi\u00f3n<\/p>\n<pre lang=\"text\"> \n   sucPerrin :: [Integer]\n<\/pre>\n<p>cuyos elementos son los n\u00fameros de Perrin. Por ejemplo,<\/p>\n<pre lang=\"text\"> \n   \u03bb> take 15 sucPerrin\n   [3,0,2,3,2,5,5,7,10,12,17,22,29,39,51]\n   \u03bb> length (show (sucPerrin !! (2*10^5)))\n   24425\n<\/pre>\n<p>Comprobar con QuickCheck si se verifica la siguiente propiedad: para todo entero n > 1, el n-\u00e9simo t\u00e9rmino de la sucesi\u00f3n de Perrin es divisible por n si y s\u00f3lo si n es primo.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericIndex, unfoldr)\nimport Data.Numbers.Primes (isPrime)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\nsucPerrin1 :: [Integer]\nsucPerrin1 = 3 : 0 : 2 : zipWith (+) sucPerrin1 (tail sucPerrin1)\n\n-- 2\u00aa soluci\u00f3n\nsucPerrin2 :: [Integer]\nsucPerrin2 = [x | (x,_,_) <- iterate op (3,0,2)]\n  where op (a,b,c) = (b,c,a+b)\n \n-- 3\u00aa soluci\u00f3n\nsucPerrin3 :: [Integer]\nsucPerrin3 =\n  unfoldr (\\(a, (b,c)) -> Just (a, (b,(c,a+b)))) (3,(0,2))\n\n-- 4\u00aa soluci\u00f3n\nsucPerrin4 :: [Integer]\nsucPerrin4 = [vectorPerrin n ! n | n <- [0..]]\n\nvectorPerrin :: Integer -> Array Integer Integer\nvectorPerrin n = v where\n  v = array (0,n) [(i,f i) | i <- [0..n]]\n  f 0 = 3\n  f 1 = 0\n  f 2 = 2\n  f i = v ! (i-2) + v ! (i-3)\n\n-- Comparaci\u00f3n de eficiencia\n--    \u03bb> length (show (sucPerrin1 !! (3*10^5)))\n--    36638\n--    (1.62 secs, 2,366,238,984 bytes)\n--    \u03bb> length (show (sucPerrin2 !! (3*10^5)))\n--    36638\n--    (1.40 secs, 2,428,701,384 bytes)\n--    \u03bb> length (show (sucPerrin3 !! (3*10^5)))\n--    36638\n--    (1.14 secs, 2,409,504,864 bytes)\n--    \u03bb> length (show (sucPerrin4 !! (3*10^5)))\n--    36638\n--    (1.78 secs, 2,585,400,776 bytes)\n\n\n-- Usaremos la 3\u00aa\nsucPerrin :: [Integer]\nsucPerrin = sucPerrin3\n\n-- La propiedad es  \nconjeturaPerrin :: Integer -> Property\nconjeturaPerrin n =\n  n > 1 ==>\n  (perrin n `mod` n == 0) == isPrime n\n\n-- (perrin n) es el n-\u00e9simo t\u00e9rmino de la sucesi\u00f3n de Perrin. Por\n-- ejemplo,\n--    perrin 4  ==  2\n--    perrin 5  ==  5\n--    perrin 6  ==  5\nperrin :: Integer -> Integer\nperrin n = sucPerrin `genericIndex` n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck conjeturaPerrin\n--    +++ OK, passed 100 tests.\n\n-- Nota: Aunque QuickCheck no haya encontrado contraejemplos, la\n-- propiedad no es cierta. S\u00f3lo lo es una de las implicaciones: si n es\n-- primo, entonces el  n-\u00e9simo t\u00e9rmino de la sucesi\u00f3n de Perrin es\n-- divisible por n. La otra es falsa y los primeros contraejemplos son\n--    271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Los n\u00fameros de Perrin se definen por la elaci\u00f3n de recurrencia P(n) = P(n &#8211; 2) + P(n &#8211; 3) si n > 2, con los valores iniciales P(0) = 3, P(1) = 0 y P(2) = 2. Definir la sucesi\u00f3n sucPerrin :: [Integer] cuyos elementos son los n\u00fameros de Perrin. Por ejemplo, \u03bb> take&#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":[8,286,174,50,89,11,6,45,146,255,76],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5904"}],"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=5904"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5904\/revisions"}],"predecessor-version":[{"id":5926,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5904\/revisions\/5926"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5904"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5904"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5904"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}