{"id":4654,"date":"2019-01-30T06:00:10","date_gmt":"2019-01-30T04:00:10","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=4654"},"modified":"2019-02-06T07:18:18","modified_gmt":"2019-02-06T05:18:18","slug":"impares-en-filas-del-triangulo-de-pascal","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/impares-en-filas-del-triangulo-de-pascal\/","title":{"rendered":"Impares en filas del tri\u00e1ngulo de Pascal"},"content":{"rendered":"<p>El tri\u00e1ngulo de Pascal es un tri\u00e1ngulo de n\u00fameros<\/p>\n<pre lang=\"text\"> \n         1\n        1 1\n       1 2 1\n     1  3 3  1\n    1 4  6  4 1\n   1 5 10 10 5 1\n  ...............\n<\/pre>\n<p>construido de la siguiente forma<\/p>\n<ul>\n<li>la primera fila est\u00e1 formada por el n\u00famero 1;<\/li>\n<li>las filas siguientes se construyen sumando los n\u00fameros adyacentes de la fila superior y a\u00f1adiendo un 1 al principio y al final de la fila. <\/li>\n<\/ul>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\"> \n   imparesPascal          :: [[Integer]]\n   nImparesPascal         :: [Int]\n   grafica_nImparesPascal :: Int -> IO ()\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>imparesPascal es la lista de los elementos impares en cada una de las filas del tri\u00e1ngulo de Pascal. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">   \n     \u03bb> take 8 imparesPascal\n     [[1],[1,1],[1,1],[1,3,3,1],[1,1],[1,5,5,1],[1,15,15,1],[1,7,21,35,35,21,7,1]]\n<\/pre>\n<ul>\n<li>nImparesPascal es la lista del n\u00famero de elementos impares en cada una de las filas del tri\u00e1ngulo de Pascal. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">   \n     \u03bb> take 32 nImparesPascal\n     [1,2,2,4,2,4,4,8,2,4,4,8,4,8,8,16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32]\n     \u03bb> maximum (take (10^6) nImparesPascal3)\n     524288\n<\/pre>\n<ul>\n<li><code>(grafica_nImparesPascal n)<\/code> dibuja la gr\u00e1fica de los n primeros t\u00e9rminos de nImparesPascal. Por ejemplo, <code>(grafica_nImparesPascal 50)<\/code> dibuja<\/li>\n<\/ul>\n<p><a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/01\/Impares_en_filas_del_triangulo_de_Pascal_50.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/01\/Impares_en_filas_del_triangulo_de_Pascal_50.png?resize=640%2C480\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-4655\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/01\/Impares_en_filas_del_triangulo_de_Pascal_50.png?w=640&amp;ssl=1 640w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/01\/Impares_en_filas_del_triangulo_de_Pascal_50.png?resize=300%2C225&amp;ssl=1 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/a><\/p>\n<p>y <code>(grafica_nImparesPascal 100)<\/code> dibuja<\/p>\n<p><a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/01\/Impares_en_filas_del_triangulo_de_Pascal_100.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/01\/Impares_en_filas_del_triangulo_de_Pascal_100.png?resize=640%2C480\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-4656\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/01\/Impares_en_filas_del_triangulo_de_Pascal_100.png?w=640&amp;ssl=1 640w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2019\/01\/Impares_en_filas_del_triangulo_de_Pascal_100.png?resize=300%2C225&amp;ssl=1 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/a><\/p>\n<p>Comprobar con QuickCheck que todos los elementos de nImparesPascal son potencias de dos.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (transpose)\nimport Test.QuickCheck\nimport Graphics.Gnuplot.Simple\n\n-- 1\u00aa definici\u00f3n de imparesPascal\n-- ==============================\n\nimparesPascal :: [[Integer]]\nimparesPascal =\n  map (filter odd) pascal\n\n-- pascal es la lista de las filas del tri\u00e1ngulo de Pascal. Por ejemplo,\n--    \u03bb> take 7 pascal\n--    [[1],[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1],[1,5,10,10,5,1],[1,6,15,20,15,6,1]]\npascal :: [[Integer]]\npascal = [1] : map f pascal\n  where f xs = zipWith (+) (0:xs) (xs++[0])\n\n-- 2\u00aa definici\u00f3n de imparesPascal\n-- ==============================\n\nimparesPascal2 :: [[Integer]]\nimparesPascal2 =\n  map (filter odd) pascal\n\npascal2 :: [[Integer]]\npascal2 = iterate f [1]\n  where f xs = zipWith (+) (0:xs) (xs++[0])\n\n-- 1\u00aa definici\u00f3n de nImparesPascal\n-- ===============================\n\nnImparesPascal :: [Int]\nnImparesPascal =\n  map length imparesPascal\n\n-- 2\u00aa definici\u00f3n de nImparesPascal\n-- ===============================\n\nnImparesPascal2 :: [Int]\nnImparesPascal2 =\n  map (length . filter odd) imparesPascal\n\n-- 3\u00aa definici\u00f3n de nImparesPascal\n-- ===============================\n\n--    \u03bb> take 32 nImparesPascal2\n--    [1,2,\n--     2,4,\n--     2,4,4,8,\n--     2,4,4,8,4,8,8,16,\n--     2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32]\nnImparesPascal3 :: [Int]\nnImparesPascal3 = 1 : zs\n  where zs = 2 : concat (transpose [zs, map (*2) zs])\n\n-- Definici\u00f3n de grafica_nImparesPascal\n-- =========================================\n\ngrafica_nImparesPascal :: Int -> IO ()\ngrafica_nImparesPascal n =\n  plotListStyle\n    [ Key Nothing\n    , PNG (\"Impares_en_filas_del_triangulo_de_Pascal_\" ++ show n ++ \".png\")\n    ]\n    (defaultStyle {plotType = LinesPoints})\n    (take n nImparesPascal3)\n\n-- Propiedad de nImparesPascal\n-- ===========================\n\n-- La propiedad es\nprop_nImparesPascal :: Positive Int -> Bool\nprop_nImparesPascal (Positive n) =\n  esPotenciaDeDos (nImparesPascal3 !! n)\n\n-- (esPotenciaDeDos n) se verifica si n es una potencia de dos. Por\n-- ejemplo,\n--    esPotenciaDeDos 16  ==  True\n--    esPotenciaDeDos 18  ==  False\nesPotenciaDeDos :: Int -> Bool\nesPotenciaDeDos 1 = True\nesPotenciaDeDos n = even n && esPotenciaDeDos (n `div` 2)\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_nImparesPascal\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nDe lo que llaman los hombres<br \/>\nvirtud, justicia y bondad,<br \/>\nuna mitad es envidia,<br \/>\ny la otra no es caridad.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>El tri\u00e1ngulo de Pascal es un tri\u00e1ngulo de n\u00fameros 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 &#8230;&#8230;&#8230;&#8230;&#8230; construido de la siguiente forma la primera fila est\u00e1 formada por el n\u00famero 1; las filas siguientes se construyen sumando los n\u00fameros adyacentes de&#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":[12,464,38,28,10,92,11,463,47,68,76],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4654"}],"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=4654"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4654\/revisions"}],"predecessor-version":[{"id":4694,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4654\/revisions\/4694"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=4654"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=4654"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=4654"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}