{"id":7402,"date":"2022-09-30T06:00:54","date_gmt":"2022-09-30T04:00:54","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7402"},"modified":"2022-12-14T14:19:26","modified_gmt":"2022-12-14T12:19:26","slug":"triangulo-aritmetico","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/triangulo-aritmetico\/","title":{"rendered":"Tri\u00e1ngulo aritm\u00e9tico"},"content":{"rendered":"<p>Los tri\u00e1ngulos aritm\u00e9ticos se forman como sigue<\/p>\n<pre lang=\"text\">\n    1\n    2  3\n    4  5  6\n    7  8  9 10\n   11 12 13 14 15\n   16 17 18 19 20 21\n<\/pre>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   linea     :: Integer -> [Integer]\n   triangulo :: Integer -> [[Integer]]\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li><code>linea n<\/code> es la l\u00ednea n-\u00e9sima de los tri\u00e1ngulos aritm\u00e9ticos. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     linea 4  ==  [7,8,9,10]\n     linea 5  ==  [11,12,13,14,15]\n     head (linea (10^20)) == 4999999999999999999950000000000000000001\n<\/pre>\n<ul>\n<li><code>triangulo n<\/code> es el tri\u00e1ngulo aritm\u00e9tico de altura <code>n<\/code>. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     triangulo 3  ==  [[1],[2,3],[4,5,6]]\n     triangulo 4  ==  [[1],[2,3],[4,5,6],[7,8,9,10]]\n<\/pre>\n<p><b>Soluciones<\/b><\/p>\n<p>A continuaci\u00f3n se muestran las <a href=\"#haskell\">soluciones en Haskell<\/a> y las <a href=\"#python\">soluciones en Python<\/a>.<\/p>\n<p><a name=\"haskell\"><\/a><br \/>\n<b>Soluciones en Haskell<\/b><\/p>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\n-- 1\u00aa definici\u00f3n de l\u00ednea\n-- ======================\n\nlinea1 :: Integer -> [Integer]\nlinea1 n = [suma1 (n-1)+1..suma1 n]\n\n-- (suma n) es la suma de los n primeros n\u00fameros. Por ejemplo,\n--    suma 3  ==  6\nsuma1 :: Integer -> Integer\nsuma1 n = sum [1..n]\n\n-- 2\u00aa definici\u00f3n de l\u00ednea\n-- ======================\n\nlinea2 :: Integer -> [Integer]\nlinea2 n = [s+1..s+n]\n  where s = suma1 (n-1)\n\n-- 3\u00aa definici\u00f3n de l\u00ednea\n-- ======================\n\nlinea3 :: Integer -> [Integer]\nlinea3 n = [s+1..s+n]\n  where s = suma2 (n-1)\n\nsuma2 :: Integer -> Integer\nsuma2 n = (1+n)*n `div` 2\n\n-- Comprobaci\u00f3n de equivalencia de linea\n-- =====================================\n\n-- La propiedad es\nprop_linea :: Positive Integer -> Bool\nprop_linea (Positive n) =\n  all (== linea1 n)\n      [linea2 n,\n       linea3 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_linea\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia de linea\n-- ==================================\n\n-- La comparaci\u00f3n es\n--    \u03bb> last (linea1 (10^7))\n--    50000005000000\n--    (5.10 secs, 3,945,159,856 bytes)\n--    \u03bb> last (linea2 (10^7))\n--    50000005000000\n--    (3.11 secs, 2,332,859,512 bytes)\n--    \u03bb> last (linea3 (10^7))\n--    50000005000000\n--    (0.16 secs, 720,559,384 bytes)\n\n-- 1\u00aa definici\u00f3n de triangulo\n-- ==========================\n\ntriangulo1 :: Integer -> [[Integer]]\ntriangulo1 n = [linea1 m | m <- [1..n]]\n\n-- 2\u00aa definici\u00f3n de triangulo\n-- ==========================\n\ntriangulo2 :: Integer -> [[Integer]]\ntriangulo2 n = [linea2 m | m <- [1..n]]\n\n-- 3\u00aa definici\u00f3n de triangulo\n-- ==========================\n\ntriangulo3 :: Integer -> [[Integer]]\ntriangulo3 n = [linea3 m | m <- [1..n]]\n\n-- Comprobaci\u00f3n de equivalencia de triangulo\n-- =========================================\n\n-- La propiedad es\nprop_triangulo :: Positive Integer -> Bool\nprop_triangulo (Positive n) =\n  all (== triangulo1 n)\n      [triangulo2 n,\n       triangulo3 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_triangulo\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia de triangulo\n-- ======================================\n\n-- La comparaci\u00f3n es\n--    \u03bb> last (last (triangulo1 (3*10^6)))\n--    4500001500000\n--    (2.25 secs, 1,735,919,184 bytes)\n--    \u03bb> last (last (triangulo2 (3*10^6)))\n--    4500001500000\n--    (1.62 secs, 1,252,238,872 bytes)\n--    \u03bb> last (last (triangulo3 (3*10^6)))\n--    4500001500000\n--    (0.79 secs, 768,558,776 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Triangulo_aritmetico.hs\">GitHub<\/a>.<\/p>\n<p><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\nfrom timeit import Timer, default_timer\nfrom hypothesis import given, strategies as st\n\n# 1\u00aa definici\u00f3n de l\u00ednea\n# ======================\n\n# suma(n) es la suma de los n primeros n\u00fameros. Por ejemplo,\n#    suma(3)  ==  6\ndef suma1(n: int) -> int:\n    return sum(range(1, n + 1))\n\ndef linea1(n: int) -> list[int]:\n    return list(range(suma1(n - 1) + 1, suma1(n) + 1))\n\n# 2\u00aa definici\u00f3n de l\u00ednea\n# ======================\n\ndef linea2(n: int) -> list[int]:\n    s = suma1(n-1)\n    return list(range(s + 1, s + n + 1))\n\n# 3\u00aa definici\u00f3n de l\u00ednea\n# ======================\n\ndef suma2(n: int) -> int:\n    return (1 + n) * n \/\/ 2\n\ndef linea3(n: int) -> list[int]:\n    s = suma2(n-1)\n    return list(range(s + 1, s + n + 1))\n\n# Comprobaci\u00f3n de equivalencia de linea\n# =====================================\n\n@given(st.integers(min_value=1, max_value=1000))\ndef test_suma(n):\n    r = linea1(n)\n    assert linea2(n) == r\n    assert linea3(n) == r\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q triangulo_aritmetico.py\n#    1 passed in 0.15s\n\n# Comparaci\u00f3n de eficiencia\n# =========================\n\ndef tiempo(e):\n    \"\"\"Tiempo (en segundos) de evaluar la expresi\u00f3n e.\"\"\"\n    t = Timer(e, \"\", default_timer, globals()).timeit(1)\n    print(f\"{t:0.2f} segundos\")\n\n# La comparaci\u00f3n es\n#    >>> tiempo('linea1(10**7)')\n#    0.53 segundos\n#    >>> tiempo('linea2(10**7)')\n#    0.40 segundos\n#    >>> tiempo('linea3(10**7)')\n#    0.29 segundos\n\n# 1\u00aa definici\u00f3n de triangulo\n# ==========================\n\ndef triangulo1(n: int) -> list[list[int]]:\n    return [linea1(m) for m in range(1, n + 1)]\n\n# 2\u00aa definici\u00f3n de triangulo\n# ==========================\n\ndef triangulo2(n: int) -> list[list[int]]:\n    return [linea2(m) for m in range(1, n + 1)]\n\n# 3\u00aa definici\u00f3n de triangulo\n# ==========================\n\ndef triangulo3(n: int) -> list[list[int]]:\n    return [linea3(m) for m in range(1, n + 1)]\n\n# Comprobaci\u00f3n de equivalencia de triangulo\n# =========================================\n\n@given(st.integers(min_value=1, max_value=1000))\ndef test_triangulo(n):\n    r = triangulo1(n)\n    assert triangulo2(n) == r\n    assert triangulo3(n) == r\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q triangulo_aritmetico.py\n#    1 passed in 3.44s\n\n# Comparaci\u00f3n de eficiencia de triangulo\n# ======================================\n#\n# La comparaci\u00f3n es\n#    >>> tiempo('triangulo1(10**4)')\n#    2.58 segundos\n#    >>> tiempo('triangulo2(10**4)')\n#    1.91 segundos\n#    >>> tiempo('triangulo3(10**4)')\n#    1.26 segundos\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium-Python\/blob\/main\/src\/triangulo_aritmetico.py\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Los tri\u00e1ngulos aritm\u00e9ticos se forman como sigue 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Definir las funciones linea :: Integer -> [Integer] triangulo :: Integer -> [[Integer]] tales que linea n es la l\u00ednea n-\u00e9sima de los tri\u00e1ngulos aritm\u00e9ticos. Por ejemplo,&#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":[581],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7402"}],"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=7402"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7402\/revisions"}],"predecessor-version":[{"id":7680,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7402\/revisions\/7680"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7402"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7402"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7402"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}