{"id":3709,"date":"2018-02-09T06:00:10","date_gmt":"2018-02-09T04:00:10","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=3709"},"modified":"2018-02-19T07:57:35","modified_gmt":"2018-02-19T05:57:35","slug":"periodos-de-fibonacci","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/periodos-de-fibonacci\/","title":{"rendered":"Per\u00edodos de Fibonacci"},"content":{"rendered":"<p>Los primeros t\u00e9rminos de la sucesi\u00f3n de Fibonacci son<\/p>\n<pre lang=\"text\">\n   0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610\n<\/pre>\n<p>Al calcular sus restos m\u00f3dulo 3 se obtiene<\/p>\n<pre lang=\"text\">\n   0,1,1,2,0,2,2,1, 0,1,1,2,0,2,2,1\n<\/pre>\n<p>Se observa que es peri\u00f3dica y su per\u00edodo es<\/p>\n<pre lang=\"text\">\n   0,1,1,2,0,2,2,1\n<\/pre>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   fibsMod                   :: Integer -> [Integer]\n   periodoFibMod             :: Integer -> [Integer]\n   longPeriodosFibMod        :: [Int]\n   graficaLongPeriodosFibMod :: Int -> IO ()\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(fibsMod n) es la lista de los t\u00e9rminos de la sucesi\u00f3n de Fibonacci m\u00f3dulo n. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     \u03bb> take 24 (fibsMod 3)\n     [0,1,1,2,0,2,2,1, 0,1,1,2,0,2,2,1, 0,1,1,2,0,2,2,1]\n     \u03bb> take 24 (fibsMod 4)\n     [0,1,1,2,3,1, 0,1,1,2,3,1, 0,1,1,2,3,1, 0,1,1,2,3,1]\n<\/pre>\n<ul>\n<li>(periodoFibMod n) es la parte perioica de la sucesi\u00f3n de Fibonacci m\u00f3dulo n. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     periodoFibMod 3  ==  [0,1,1,2,0,2,2,1]\n     periodoFibMod 4  ==  [0,1,1,2,3,1]\n     periodoFibMod 7  ==  [0,1,1,2,3,5,1,6,0,6,6,5,4,2,6,1]\n<\/pre>\n<ul>\n<li>longPeriodosFibMod es la sucesi\u00f3n de las longitudes de los per\u00edodos de las sucesiones de Fibonacci m\u00f3dulo n, para n > 0. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     \u03bb> take 20 longPeriodosFibMod\n     [1,3,8,6,20,24,16,12,24,60,10,24,28,48,40,24,36,24,18,60]\n<\/pre>\n<ul>\n<li>(graficaLongPeriodosFibMod n) dibuja la gr\u00e1fica de los n primeros t\u00e9rminos de la sucesi\u00f3n longPeriodosFibMod. Por ejemplo, (graficaLongPeriodosFibMod n) dibuja<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/02\/Periodos_de_Fibonacci-300.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/02\/Periodos_de_Fibonacci-300.png?resize=640%2C480\" alt=\"Periodos_de_Fibonacci 300\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-3710\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/02\/Periodos_de_Fibonacci-300.png?w=640&amp;ssl=1 640w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/02\/Periodos_de_Fibonacci-300.png?resize=300%2C225&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/02\/Periodos_de_Fibonacci-300.png?resize=100%2C75&amp;ssl=1 100w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/02\/Periodos_de_Fibonacci-300.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\nfibsMod :: Integer -> [Integer]\nfibsMod n = map (`mod` n) fibs\n\n-- fibs es la sucesi\u00f3n de Fibonacci. Por ejemplo,\n--    \u03bb> take 20 fibs\n--    [0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181]\nfibs :: [Integer]\nfibs = 0 : 1 : zipWith (+) fibs (tail fibs)\n\nperiodoFibMod :: Integer -> [Integer]\nperiodoFibMod 1 = [0]\nperiodoFibMod n = 0 : 1 : aux (drop 2 (fibsMod n))\n  where aux (0:1:xs) = []\n        aux (a:b:xs) = a : aux (b:xs)\n\nlongPeriodosFibMod :: [Int]\nlongPeriodosFibMod =\n  [length (periodoFibMod n) | n <- [1..]]\n\n-- 2\u00aa definici\u00f3n de longPeriodosFibMod\n-- ===================================\n\nlongPeriodosFibMod2 :: [Int]\nlongPeriodosFibMod2 = map longPeriodoFibMod [1..]\n\nlongPeriodoFibMod :: Integer -> Int\nlongPeriodoFibMod 1 = 1\nlongPeriodoFibMod n = aux 1 (tail (fibsMod n)) 0\n  where aux 0 (1 : xs) k = k\n        aux _ (x : xs) k = aux x xs (k + 1)\n\ngraficaLongPeriodosFibMod :: Int -> IO ()\ngraficaLongPeriodosFibMod n =\n  plotList [ Key Nothing\n           , Title (\"graficaLongPeriodosFibMod \" ++ show n)\n           , PNG (\"Periodos_de_Fibonacci \" ++ show n ++ \".png\")]\n           (take n longPeriodosFibMod)\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Los primeros t\u00e9rminos de la sucesi\u00f3n de Fibonacci son 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610 Al calcular sus restos m\u00f3dulo 3 se obtiene 0,1,1,2,0,2,2,1, 0,1,1,2,0,2,2,1 Se observa que es peri\u00f3dica y su per\u00edodo es 0,1,1,2,0,2,2,1 Definir las funciones fibsMod :: Integer -> [Integer] periodoFibMod :: Integer -> [Integer] longPeriodosFibMod :: [Int] graficaLongPeriodosFibMod :: Int -> IO () tales que (fibsMod&#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":[8,46,376,28,415,10,89,11,309,6,45,47,76],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3709"}],"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=3709"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3709\/revisions"}],"predecessor-version":[{"id":3772,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3709\/revisions\/3772"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3709"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3709"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3709"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}