{"id":5073,"date":"2019-06-10T06:00:14","date_gmt":"2019-06-10T04:00:14","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5073"},"modified":"2021-04-25T12:38:50","modified_gmt":"2021-04-25T10:38:50","slug":"numeros-de-perrin-2019","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/numeros-de-perrin-2019\/","title":{"rendered":"N\u00fameros de Perrin"},"content":{"rendered":"

Los n\u00fameros de Perrin<\/a> se definen por la elaci\u00f3n de recurrencia<\/p>\n

 \n   P(n) = P(n - 2) + P(n - 3) si n > 2,\n<\/pre>\n
con los valores iniciales<\/p>\n
 \n   P(0) = 3, P(1) = 0 y P(2) = 2.\n<\/pre>\n
Definir la sucesi\u00f3n<\/p>\n
 \n   sucPerrin :: [Integer]\n<\/pre>\n
cuyos elementos son los n\u00fameros de Perrin. Por ejemplo,<\/p>\n
 \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
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
Soluciones<\/h4>\n
[schedule expon=’2019-06-12′ expat=\u00bb06:00″]<\/p>\n
    \n
  • Las soluciones se pueden escribir en los comentarios hasta el 12 de junio.\n
  • El c\u00f3digo se debe escribir entre una l\u00ednea con <pre lang=\u00bbhaskell\u00bb> y otra con <\/pre>\n<\/ul>\n
    Pensamiento<\/h4>\n
    \nEncuentro lo que no busco:
    \nlas hojas del toronjil
    \nhuelen a lim\u00f3n maduro.<\/p>\n
    Antonio Machado\n<\/p><\/blockquote>\n
    [\/schedule]<\/p>\n
    [schedule on=’2019-06-12′ at=\u00bb06:00″]<\/p>\n
    \r\nimport Data.List (genericIndex, unfoldr)\r\nimport Data.Numbers.Primes (isPrime)\r\nimport Test.QuickCheck\r\n\r\n-- 1\u00aa soluci\u00f3n\r\nsucPerrin1 :: [Integer]\r\nsucPerrin1 = 3 : 0 : 2 : zipWith (+) sucPerrin1 (tail sucPerrin1)\r\n\r\n-- 2\u00aa soluci\u00f3n\r\nsucPerrin2 :: [Integer]\r\nsucPerrin2 = [x | (x,_,_) <- iterate op (3,0,2)]\r\n  where op (a,b,c) = (b,c,a+b)\r\n \r\n-- 3\u00aa soluci\u00f3n\r\nsucPerrin3 :: [Integer]\r\nsucPerrin3 =\r\n  unfoldr (\\(a, (b,c)) -> Just (a, (b,(c,a+b)))) (3,(0,2))\r\n\r\n-- Comparaci\u00f3n de eficiencia\r\n--    \u03bb> length (show (sucPerrin1 !! (10^5)))\r\n--    12213\r\n--    (1.44 secs, 295,373,680 bytes)\r\n--    \u03bb> length (show (sucPerrin2 !! (10^5)))\r\n--    12213\r\n--    (1.22 secs, 301,493,408 bytes)\r\n--    \u03bb> length (show (sucPerrin3 !! (10^5)))\r\n--    12213\r\n--    (0.86 secs, 296,911,304 bytes)\r\n\r\n-- Usaremos la 3\u00aa\r\nsucPerrin :: [Integer]\r\nsucPerrin = sucPerrin3\r\n\r\n-- La propiedad es  \r\nconjeturaPerrin :: Integer -> Property\r\nconjeturaPerrin n =\r\n  n > 1 ==>\r\n  (perrin n `mod` n == 0) == isPrime n\r\n\r\n-- (perrin n) es el n-\u00e9simo t\u00e9rmino de la sucesi\u00f3n de Perrin. Por\r\n-- ejemplo,\r\n--    perrin 4  ==  2\r\n--    perrin 5  ==  5\r\n--    perrin 6  ==  5\r\nperrin :: Integer -> Integer\r\nperrin n = sucPerrin `genericIndex` n\r\n\r\n-- La comprobaci\u00f3n es\r\n--    \u03bb> quickCheck conjeturaPerrin\r\n--    +++ OK, passed 100 tests.\r\n\r\n-- Nota: Aunque QuickCheck no haya encontrado contraejemplos, la\r\n-- propiedad no es cierta. S\u00f3lo lo es una de las implicaciones: si n es\r\n-- primo, entonces el  n-\u00e9simo t\u00e9rmino de la sucesi\u00f3n de Perrin es\r\n-- divisible por n. La otra es falsa y los primeros contraejemplos son\r\n--    271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291\r\n<\/pre>\n
    [\/schedule]<\/p>\n","protected":false},"excerpt":{"rendered":"
    Los n\u00fameros de Perrin se definen por la elaci\u00f3n de recurrencia P(n) = P(n – 2) + P(n – 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…<\/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":[2],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5073"}],"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=5073"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5073\/revisions"}],"predecessor-version":[{"id":5074,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5073\/revisions\/5074"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5073"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5073"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5073"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}