Definir la funci\u00f3n<\/p>\n
\n sumas3Capicuas :: Integer -> [(Integer, Integer, Integer)]\n<\/pre>\ntales que (sumas3Capicuas x) es la lista de las descomposiciones de x como suma de tres capic\u00faas (con los sumandos no decrecientes). Por ejemplo,<\/p>\n
\n sumas3Capicuas 0 == [(0,0,0)]\n sumas3Capicuas 1 == [(0,0,1)]\n sumas3Capicuas 2 == [(0,0,2),(0,1,1)]\n sumas3Capicuas 3 == [(0,0,3),(0,1,2),(1,1,1)]\n sumas3Capicuas 4 == [(0,0,4),(0,1,3),(0,2,2),(1,1,2)]\n length (sumas3Capicuas 17) == 17\n length (sumas3Capicuas 2017) == 47\n length (sumas3Capicuas 999999) == 15266\n<\/pre>\nComprobar con QuickCheck que todo n\u00famero natural se puede escribir como suma de tres capic\u00faas.<\/p>\n
Soluciones<\/h4>\n\nimport Test.QuickCheck\n\nsumas3Capicuas :: Integer -> [(Integer, Integer, Integer)]\nsumas3Capicuas x =\n [(a,b,c) | a <- as\n , b <- dropWhile (< a) as\n , let c = x - a - b\n , b <= c \n , esCapicua c]\n where as = takeWhile (<= x) capicuas\n\n-- capicuas es la sucesi\u00f3n de los n\u00fameros capic\u00faas. Por ejemplo,\n-- \u03bb> take 45 capicuas\n-- [0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99,101,111,121,131,\n-- 141,151,161,171,181,191,202,212,222,232,242,252,262,272,282,292,\n-- 303,313,323,333,343,353]\n-- Se usar\u00e1 la 2\u00aa definici\u00f3n del ejercicio \"Sucesi\u00f3n de capic\u00faas\".\ncapicuas :: [Integer]\ncapicuas = capicuasImpares `mezcla` capicuasPares\n\n-- capicuasPares es la sucesi\u00f3n del cero y las capic\u00faas con un n\u00famero\n-- par de d\u00edgitos. Por ejemplo, \n-- \u03bb> take 17 capicuasPares\n-- [0,11,22,33,44,55,66,77,88,99,1001,1111,1221,1331,1441,1551,1661]\ncapicuasPares :: [Integer]\ncapicuasPares =\n [read (ns ++ reverse ns) | n <- [0..]\n , let ns = show n] \n\n-- capicuasImpares es la sucesi\u00f3n de las capic\u00faas con un n\u00famero\n-- impar de d\u00edgitos a partir de 1. Por ejemplo, \n-- \u03bb> take 20 capicuasImpares\n-- [1,2,3,4,5,6,7,8,9,101,111,121,131,141,151,161,171,181,191,202]\ncapicuasImpares :: [Integer]\ncapicuasImpares =\n [1..9] ++ [read (ns ++ [z] ++ reverse ns)\n | n <- [1..]\n , let ns = show n\n , z <- \"0123456789\"] \n\n-- (mezcla xs ys) es la lista ordenada obtenida mezclando las dos listas\n-- ordenadas xs e ys, suponiendo que ambas son infinitas y con elementos\n-- distintos. Por ejemplo,\n-- take 10 (mezcla [2,12..] [5,15..]) == [2,5,12,15,22,25,32,35,42,45]\n-- take 10 (mezcla [2,22..] [5,15..]) == [2,5,15,22,25,35,42,45,55,62]\nmezcla :: Ord a => [a] -> [a] -> [a]\nmezcla us@(x:xs) vs@(y:ys)\n | x < y = x : mezcla xs vs\n | otherwise = y : mezcla us ys\n\n-- (esCapicua x) se verifica si x es capic\u00faa. Por ejemplo,\n-- esCapicua 353 == True\n-- esCapicua 3553 == True\n-- esCapicua 3535 == False\nesCapicua :: Integer -> Bool\nesCapicua x =\n xs == reverse xs\n where xs = show x\n\n-- La propiedad es\nprop_sumas3Capicuas :: Integer -> Property\nprop_sumas3Capicuas x =\n x >= 0 ==> not (null (sumas3Capicuas x))\n\n-- La comprobaci\u00f3n es\n-- \u03bb> quickCheck prop_sumas3Capicuas\n-- +++ OK, passed 100 tests.\n<\/pre>\n","protected":false},"excerpt":{"rendered":"Definir la funci\u00f3n sumas3Capicuas :: Integer -> [(Integer, Integer, Integer)] tales que (sumas3Capicuas x) es la lista de las descomposiciones de x como suma de tres capic\u00faas (con los sumandos no decrecientes). Por ejemplo, sumas3Capicuas 0 == [(0,0,0)] sumas3Capicuas 1 == [(0,0,1)] sumas3Capicuas 2 == [(0,0,2),(0,1,1)] sumas3Capicuas 3 == [(0,0,3),(0,1,2),(1,1,1)] sumas3Capicuas 4 == [(0,0,4),(0,1,3),(0,2,2),(1,1,2)] length…<\/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":[8,59,11,95,6,32,33,34,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2802"}],"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=2802"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2802\/revisions"}],"predecessor-version":[{"id":2840,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2802\/revisions\/2840"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=2802"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=2802"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=2802"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}