Los n\u00fameros triangulares se forman como sigue<\/p>\n
\n * * *\n * * * *\n * * *\n 1 3 6\n<\/pre>\nLa sucesi\u00f3n de los n\u00fameros triangulares se obtiene sumando los n\u00fameros naturales. As\u00ed, los 5 primeros n\u00fameros triangulares son<\/p>\n
\n 1 = 1\n 3 = 1 + 2\n 6 = 1 + 2 + 3\n 10 = 1 + 2 + 3 + 4\n 15 = 1 + 2 + 3 + 4 + 5\n<\/pre>\nDefinir la funci\u00f3n<\/p>\n
\n descomposicionesTriangulares :: Int -> [(Int, Int, Int)]\n<\/pre>\ntal que (descomposicionesTriangulares n)<\/code> es la lista de las ternas correspondientes a las descomposiciones de n en tres sumandos formados por n\u00fameros triangulares. Por ejemplo,<\/p>\n\n descomposicionesTriangulares 4 == []\n descomposicionesTriangulares 5 == [(1,1,3)]\n descomposicionesTriangulares 12 == [(1,1,10),(3,3,6)]\n descomposicionesTriangulares 30 == [(1,1,28),(3,6,21),(10,10,10)]\n descomposicionesTriangulares 61 == [(1,15,45),(3,3,55),(6,10,45),(10,15,36)]\n descomposicionesTriangulares 52 == [(1,6,45),(1,15,36),(3,21,28),(6,10,36),(10,21,21)]\n descomposicionesTriangulares 82 == [(1,3,78),(1,15,66),(1,36,45),(6,10,66),(6,21,55),(10,36,36)]\n length (descomposicionesTriangulares (5*10^5)) == 124\n<\/pre>\nSoluciones<\/h4>\n
[schedule expon=’2022-04-20′ expat=\u00bb06:00″]<\/p>\n
\n- Las soluciones se pueden escribir en los comentarios.\n
- El c\u00f3digo se debe escribir entre una l\u00ednea con <pre lang="haskell"> y otra con <\/pre>\n<\/ul>\n
[\/schedule]<\/p>\n
[schedule on=’2022-04-20′ at=\u00bb06:00″]<\/p>\n
\r\nimport Test.QuickCheck\r\n\r\n-- 1\u00aa soluci\u00f3n\r\n-- ===========\r\n\r\ndescomposicionesTriangulares1 :: Int -> [(Int, Int, Int)]\r\ndescomposicionesTriangulares1 n =\r\n [(x,y,z) | x <- xs,\r\n y <- xs,\r\n z <- xs,\r\n x <= y && y <= z,\r\n x + y + z == n]\r\n where xs = takeWhile (<=n) triangulares\r\n\r\n-- triangulares es la lista de los n\u00fameros triangulares. Por ejemplo,\r\n-- take 9 triangulares == [1,3,6,10,15,21,28,36,45]\r\ntriangulares :: [Int]\r\ntriangulares = scanl (+) 1 [2..]\r\n\r\n-- 2\u00aa soluci\u00f3n\r\n-- ===========\r\n\r\ndescomposicionesTriangulares2 :: Int -> [(Int, Int, Int)]\r\ndescomposicionesTriangulares2 n =\r\n [(x,y,z) | x <- xs,\r\n y <- xs,\r\n x <= y,\r\n z <- xs,\r\n y <= z,\r\n x + y + z == n]\r\n where xs = takeWhile (<=n) triangulares\r\n\r\n-- 3\u00aa soluci\u00f3n\r\n-- ===========\r\n\r\ndescomposicionesTriangulares3 :: Int -> [(Int, Int, Int)]\r\ndescomposicionesTriangulares3 n =\r\n [(x,y,z) | x <- xs,\r\n y <- xs,\r\n x <= y,\r\n let z = n - x - y,\r\n y <= z,\r\n z `elem` xs]\r\n where xs = takeWhile (<=n) triangulares\r\n\r\n-- 4\u00aa soluci\u00f3n\r\n-- ===========\r\n\r\ndescomposicionesTriangulares4 :: Int -> [(Int, Int, Int)]\r\ndescomposicionesTriangulares4 n =\r\n [(x,y,n-x-y) | x <- xs,\r\n y <- dropWhile ( Bool\r\nprop_descomposicionesTriangulares (Positive n) =\r\n all (== descomposicionesTriangulares1 n)\r\n [descomposicionesTriangulares2 n,\r\n descomposicionesTriangulares3 n,\r\n descomposicionesTriangulares4 n]\r\n\r\n-- La comprobaci\u00f3n es\r\n-- \u03bb> quickCheck prop_descomposicionesTriangulares\r\n-- +++ OK, passed 100 tests.\r\n\r\n-- Comparaci\u00f3n de eficiencia\r\n-- =========================\r\n\r\n-- La comparaci\u00f3n es\r\n-- \u03bb> last (descomposicionesTriangulares1 (2*10^4))\r\n-- (5671,6328,8001)\r\n-- (3.34 secs, 1,469,517,168 bytes)\r\n-- \u03bb> last (descomposicionesTriangulares2 (2*10^4))\r\n-- (5671,6328,8001)\r\n-- (1.29 secs, 461,433,928 bytes)\r\n-- \u03bb> last (descomposicionesTriangulares3 (2*10^4))\r\n-- (5671,6328,8001)\r\n-- (0.08 secs, 6,574,056 bytes)\r\n--\r\n-- \u03bb> last (descomposicionesTriangulares3 (5*10^5))\r\n-- (140185,148240,211575)\r\n-- (2.12 secs, 151,137,280 bytes)\r\n-- \u03bb> last (descomposicionesTriangulares4 (5*10^5))\r\n-- (140185,148240,211575)\r\n-- (2.30 secs, 103,280,216 bytes)\r\n<\/pre>\nEl c\u00f3digo se encuentra en [GitHub](https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Descomposiciones_triangulares.hs).<\/p>\n
La elaboraci\u00f3n de las soluciones se describe en el siguiente v\u00eddeo<\/p>\n
Nuevas soluciones<\/h4>\n\n- En los comentarios se pueden escribir nuevas soluciones.\n
- El c\u00f3digo se debe escribir entre una l\u00ednea con <pre lang="haskell"> y otra con <\/pre>\n<\/ul>\n
[\/schedule]<\/p>\n","protected":false},"excerpt":{"rendered":"
Los n\u00fameros triangulares se forman como sigue * * * * * * * * * * 1 3 6 La sucesi\u00f3n de los n\u00fameros triangulares se obtiene sumando los n\u00fameros naturales. As\u00ed, los 5 primeros n\u00fameros triangulares son 1 = 1 3 = 1 + 2 6 = 1 + 2 + 3 10…<\/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":[41,8,26,415,11,78,34,521,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6901"}],"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=6901"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6901\/revisions"}],"predecessor-version":[{"id":6902,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6901\/revisions\/6902"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6901"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6901"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6901"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}