{"id":3326,"date":"2017-05-23T06:00:18","date_gmt":"2017-05-23T04:00:18","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=3326"},"modified":"2022-03-26T14:23:59","modified_gmt":"2022-03-26T12:23:59","slug":"polinomios-de-fibonacci-17","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/polinomios-de-fibonacci-17\/","title":{"rendered":"Polinomios de Fibonacci"},"content":{"rendered":"<p>La sucesi\u00f3n de <a href=\"https:\/\/en.wikipedia.org\/wiki\/Fibonacci_polynomials\">polinomios de Fibonacci<\/a> se define por<\/p>\n<pre lang=\"text\">\n   p(0) = 0\n   p(1) = 1\n   p(n) = x*p(n-1) + p(n-2)\n<\/pre>\n<p>Los primeros t\u00e9rminos de la sucesi\u00f3n son<\/p>\n<pre lang=\"text\">\n   p(2) = x\n   p(3) = x^2 + 1\n   p(4) = x^3 + 2*x\n   p(5) = x^4 + 3*x^2 + 1\n<\/pre>\n<p>Definir la lista<\/p>\n<pre lang=\"text\">\n   sucPolFib :: [Polinomio Integer]\n<\/pre>\n<p>tal que sus elementos son los polinomios de Fibonacci. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> take 7 sucPolFib\n   [0,1,1*x,x^2 + 1,x^3 + 2*x,x^4 + 3*x^2 + 1,x^5 + 4*x^3 + 3*x]\n   \u03bb> sum (map grado (take 3000 sucPolFib2))\n   4495501\n<\/pre>\n<p>Comprobar con QuickCheck que el valor del n-\u00e9simo t\u00e9rmino de sucPolFib para x=1 es el n-\u00e9simo t\u00e9rmino de la sucesi\u00f3n de Fibonacci 0, 1, 1, 2, 3, 5, 8, &#8230;<\/p>\n<p><strong>Nota<\/strong>. Limitar la b\u00fasqueda a ejemplos peque\u00f1os usando<\/p>\n<pre lang=\"text\">\n   quickCheckWith (stdArgs {maxSize=5}) prop_polFib\n<\/pre>\n<h4>Soluciones<\/h4>\n<p>[schedule expon=&#8217;2017-05-30&#8242; expat=\u00bb06:00&#8243;]<\/p>\n<ul>\n<li>Las soluciones se pueden escribir en los comentarios hasta el 30 de mayo.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=\u00bbhaskell\u00bb&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n<p>[\/schedule]<\/p>\n<p>[schedule on=&#8217;2017-05-30&#8242; at=\u00bb06:00&#8243;]<\/p>\n<pre lang=\"haskell\">\r\nimport Data.List (genericIndex)\r\nimport I1M.PolOperaciones\r\nimport Test.QuickCheck\r\n\r\n-- 1\u00aa soluci\u00f3n\r\n-- ===========\r\n\r\nsucPolFib :: [Polinomio Integer]\r\nsucPolFib = [polFibR n | n <- [0..]]\r\n\r\npolFibR :: Integer -> Polinomio Integer\r\npolFibR 0 = polCero\r\npolFibR 1 = polUnidad\r\npolFibR n = \r\n  sumaPol (multPol (consPol 1 1 polCero) (polFibR (n-1)))\r\n          (polFibR (n-2))\r\n\r\n-- 2\u00aa definici\u00f3n (din\u00e1mica)\r\n-- ========================\r\n\r\nsucPolFib2 :: [Polinomio Integer]\r\nsucPolFib2 = \r\n  polCero : polUnidad : zipWith f (tail sucPolFib2) sucPolFib2\r\n  where f p = sumaPol (multPol (consPol 1 1 polCero) p)\r\n\r\n-- La propiedad es\r\nprop_polFib :: Integer -> Property\r\nprop_polFib n = \r\n    n >= 0 ==> valor (polFib n) 1 == fib n\r\n    where polFib n = sucPolFib2 `genericIndex` n\r\n          fib n    = fibs `genericIndex` n\r\n\r\nfibs :: [Integer]\r\nfibs = 0 : 1 : zipWith (+) fibs (tail fibs)\r\n\r\n-- La comprobaci\u00f3n es\r\n--    ghci> quickCheckWith (stdArgs {maxSize=5}) prop_polFib\r\n--    +++ OK, passed 100 tests.\r\n<\/pre>\n<p>[\/schedule]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>La sucesi\u00f3n de polinomios de Fibonacci se define por p(0) = 0 p(1) = 1 p(n) = x*p(n-1) + p(n-2) Los primeros t\u00e9rminos de la sucesi\u00f3n son p(2) = x p(3) = x^2 + 1 p(4) = x^3 + 2*x p(5) = x^4 + 3*x^2 + 1 Definir la lista sucPolFib :: [Polinomio Integer] tal&#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":[2],"tags":[265],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3326"}],"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=3326"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3326\/revisions"}],"predecessor-version":[{"id":3329,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3326\/revisions\/3329"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3326"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3326"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3326"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}