{"id":5279,"date":"2019-12-23T05:30:48","date_gmt":"2019-12-23T03:30:48","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5279"},"modified":"2019-12-30T09:20:25","modified_gmt":"2019-12-30T07:20:25","slug":"suma-de-numeros-de-fibonacci-con-indice-impar","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/suma-de-numeros-de-fibonacci-con-indice-impar\/","title":{"rendered":"Suma de n\u00fameros de Fibonacci con \u00edndice impar"},"content":{"rendered":"<p>La sucesi\u00f3n de Fibonacci, F(n), es la siguiente sucesi\u00f3n infinita de n\u00fameros naturales:<\/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>La sucesi\u00f3n comienza con los n\u00fameros 0 y 1. A partir de estos, cada t\u00e9rmino es la suma de los dos anteriores.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   sumaFibsIndiceImpar :: Int -> Integer\n<\/pre>\n<p>tal que (sumaFibsIndiceImpar n) es la suma de los n primeros t\u00e9rminos de la sucesi\u00f3n de Fibonacci no \u00edndice impar; es decir,<\/p>\n<pre lang=\"text\">\n   sumaFibsIndiceImpar n = F(1) + F(3) + ... + F(2*n-1)\n<\/pre>\n<p>Por ejemplo,<\/p>\n<pre lang=\"text\">\n   sumaFibsIndiceImpar 1  ==  1\n   sumaFibsIndiceImpar 2  ==  3\n   sumaFibsIndiceImpar 3  ==  8\n   sumaFibsIndiceImpar 4  ==  21\n   sumaFibsIndiceImpar 5  ==  55\n   sumaFibsIndiceImpar (10^4) `rem` (10^9)  ==  213093125\n<\/pre>\n<p>En los ejemplos anteriores se observa que<\/p>\n<pre lang=\"text\">\n   sumaFibsIndiceImpar 1  ==  F(2)\n   sumaFibsIndiceImpar 2  ==  F(4)\n   sumaFibsIndiceImpar 3  ==  F(6)\n   sumaFibsIndiceImpar 4  ==  F(8)\n   sumaFibsIndiceImpar 5  ==  F(10)\n<\/pre>\n<p>Comprobar con QuickCheck que (sumaFibsIndiceImpar n) es F(2n); es decir, el 2n-\u00e9simo n\u00famero de Fibonacci<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsumaFibsIndiceImpar :: Int -> Integer\nsumaFibsIndiceImpar n = sum [fib (2*k-1) | k <- [1..n]]\n\n-- (fib n) es el n-\u00e9simo t\u00e9rmino de la sucesi\u00f3n de Fibonacci. Por\n-- ejemplo,\n--    fib 6  ==  8\nfib :: Int -> Integer\nfib n = fibs !! n\n\n-- fibs es la lista de t\u00e9rminos de 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\n-- 2\u00aa soluci\u00f3n\n-- ============\n\nsumaFibsIndiceImpar2 :: Int -> Integer\nsumaFibsIndiceImpar2 n =\n  sum [a | (a,b) <- zip fibs [0..2*n], odd b]\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> sumaFibsIndiceImpar (10^4) `rem` (10^9)\n--    213093125\n--    (0.98 secs, 13,889,312 bytes)\n--    \u03bb> sumaFibsIndiceImpar2 (10^4) `rem` (10^9)\n--    213093125\n--    (0.05 secs, 18,047,720 bytes)\n\n-- Comprobaci\u00f3n\n-- ============\n\n-- La propiedad es\nprop_sumaFibsIndiceImpar :: Int -> Property\nprop_sumaFibsIndiceImpar n =\n  n >= 0 ==> sumaFibsIndiceImpar n == fib (2*n)\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_sumaFibsIndiceImpar\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Referencia<\/h4>\n<ul>\n<li><a href=\"http:\/\/bit.ly\/2Q9bvoP\">Sum of sequence of odd index Fibonacci numbers<\/a> en ProofWiki.<\/li>\n<\/ul>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nEl coraz\u00f3n del poeta, tan rico en sonoridades, es casi un insulto a la afon\u00eda cordial de la masa.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>La sucesi\u00f3n de Fibonacci, F(n), es la siguiente sucesi\u00f3n infinita de n\u00fameros naturales: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, &#8230; La sucesi\u00f3n comienza con los n\u00fameros 0 y 1. A partir de estos, cada t\u00e9rmino es la suma de los dos anteriores. Definir la funci\u00f3n&#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":[8,415,11,40,45,146,76],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5279"}],"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=5279"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5279\/revisions"}],"predecessor-version":[{"id":5324,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5279\/revisions\/5324"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5279"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5279"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5279"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}