Representación binaria de los números de Carol

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1. Haskell

   import Test.QuickCheck carol :: Integer -> Integer carol n = (2^n-1)^2 - 2 carolBinario :: Integer -> Integer carolBinario n = bin (carol n) bin :: Integer -> Integer bin n = read (concat [show a | a <- dec2bin n [] ]) dec2bin :: Integer -> [Integer] -> [Integer] dec2bin 0 xs = xs dec2bin n xs = dec2bin (n `div` 2) ((n `mod` 2):xs) prop_carol :: Integer -> Property prop_carol n = n > 2 ==> carolBinario n == read (concat [show a | a <- (replica (n-2) 1++[0]++replica (n+1) 1)]) -- *Main> quickCheck prop_carol -- +++ OK, passed 100 tests. replica :: Integral i => i -> a -> [a] replica 0 _ = [] replica n a = a : replica (n-1) a

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   import Test.QuickCheck   carol :: Integer -> Integer carol n = (2^n-1)^2 - 2   carolBinario :: Integer -> Integer carolBinario n = bin (carol n)   bin :: Integer -> Integer bin n = read (concat [show a | a <- dec2bin n [] ])   dec2bin :: Integer -> [Integer] -> [Integer] dec2bin 0 xs = xs dec2bin n xs = dec2bin (n `div` 2) ((n `mod` 2):xs)   prop_carol :: Integer -> Property prop_carol n = n > 2 ==>   carolBinario n ==   read (concat [show a | a <- (replica (n-2) 1++[0]++replica (n+1) 1)])   --    *Main> quickCheck prop_carol --    +++ OK, passed 100 tests.   replica :: Integral i => i -> a -> [a] replica 0 _ = [] replica n a = a : replica (n-1) a

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2. Haskell

   carol :: Integer -> Integer carol n = (2^n-1)^2 - 2 -- Definicion directa carolBinario :: Integer -> Integer carolBinario = read . concat . map show . reverse . aBinario. carol aBinario :: Integer -> [Integer] aBinario n | n < 2 = [n] | otherwise = n `mod` 2 : aBinario (div n 2) -- Definición usando la propiedad, esta se puede probar de forma de -- forma directa operarando con números binarios. carolBinario2 :: Integer -> Integer carolBinario2 n = read $ concat $ replicate (fromInteger n-2) "1" ++ ["0"] ++ replicate (fromInteger n+1) "1" prop_carol :: Integer -> Property prop_carol n = n > 2 ==> carolBinario n == carolBinario2 n

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   carol :: Integer -> Integer carol n = (2^n-1)^2 - 2   -- Definicion directa carolBinario :: Integer -> Integer carolBinario = read . concat . map show . reverse . aBinario. carol   aBinario :: Integer -> [Integer] aBinario n | n < 2     = [n]            | otherwise = n `mod` 2 : aBinario (div n 2)   -- Definición usando la propiedad, esta se puede probar de forma de -- forma directa operarando con números binarios. carolBinario2 :: Integer -> Integer carolBinario2 n =     read   $ concat   $ replicate (fromInteger n-2) "1" ++ ["0"] ++ replicate (fromInteger n+1) "1"   prop_carol :: Integer -> Property prop_carol n = n > 2 ==> carolBinario n == carolBinario2 n

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3. Haskell

   carol :: Integer -> Integer carol n = (2^n - 1)^2 - 2 carolBinario :: Integer -> Integer carolBinario n = sum (zipWith (*) (int2bin (carol n)) (iterate (*10) 1)) int2bin :: Integer -> [Integer] int2bin n | n < 2 = [n] | otherwise = mod n 2 : int2bin (div n 2) digitos :: Integer -> [Integer] digitos x = [read [d] | d <- show x] prop_Binario :: Integer -> Property prop_Binario n = n > 2 ==> digitos1 (carolBinario n) == genericTake (n-2) (repeat 1) ++ [0]++ genericTake (n+1) (repeat 1)

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   carol :: Integer -> Integer carol n = (2^n - 1)^2 - 2   carolBinario :: Integer -> Integer carolBinario n = sum (zipWith (*) (int2bin (carol n)) (iterate (*10) 1)) int2bin :: Integer -> [Integer] int2bin n | n < 2 = [n]           | otherwise = mod n 2 : int2bin (div n 2)   digitos :: Integer -> [Integer] digitos x = [read [d] | d <- show x]   prop_Binario :: Integer -> Property prop_Binario n = n > 2 ==> digitos1 (carolBinario n) == genericTake (n-2) (repeat 1) ++ [0]++ genericTake (n+1) (repeat 1)

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4. Haskell

   import Test.QuickCheck carol :: Integer -> Integer carol x = head [n | (n,m) <- zip caroles [1..], m == x] where caroles = [4^n - 2^(n+1) - 1 | n <- [1..]] carolBinario :: Integer -> Integer carolBinario = suma.aux.carol aux :: Integer -> [Integer] aux x | x < 2 = [x] | otherwise = mod x 2 : aux (div x 2) suma :: [Integer] -> Integer suma xs = sum [x*10^n | (x,n) <- zip xs [0..]] prop :: Integer -> Property prop x = x > 2 ==> carolBinarioA2 x == suma (concat [aux2 (x+1),[0],aux2 (x-2)]) aux2 :: Integer -> [Integer] aux2 n = [a | (a,b) <- zip [1,1..] [1..n]]

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   import Test.QuickCheck   carol :: Integer -> Integer carol x = head [n | (n,m) <- zip caroles [1..], m == x]   where caroles = [4^n - 2^(n+1) - 1 | n <- [1..]]   carolBinario :: Integer -> Integer carolBinario = suma.aux.carol   aux :: Integer -> [Integer] aux x | x < 2     = [x]       | otherwise = mod x 2 : aux (div x 2)   suma :: [Integer] -> Integer suma xs = sum [x*10^n | (x,n) <- zip xs [0..]]   prop :: Integer -> Property prop x =   x > 2 ==>   carolBinarioA2 x == suma (concat [aux2 (x+1),[0],aux2 (x-2)])   aux2 :: Integer -> [Integer] aux2 n = [a | (a,b) <- zip [1,1..] [1..n]]

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5. Haskell

   carol :: Int -> Int carol x = [4^n-2^(n+1)-1 | n <- [1..]] !! (x-1) carolBinario :: Int -> Int carolBinario x = binarioTotal (reverse (int2bin (carol x))) int2bin :: Int -> [Int] int2bin a | a < 2 = [a] | otherwise = mod a 2 : int2bin (div a 2) binarioTotal :: [Int] -> Int binarioTotal [x] = x binarioTotal (x:xs) = x*10^(length (x:xs)-1) + binarioTotal (xs)

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   carol :: Int -> Int carol x = [4^n-2^(n+1)-1 | n <- [1..]] !! (x-1)   carolBinario :: Int -> Int carolBinario x = binarioTotal (reverse (int2bin (carol x)))   int2bin :: Int -> [Int] int2bin a | a < 2 = [a]           | otherwise = mod a 2 : int2bin (div a 2)   binarioTotal :: [Int] -> Int binarioTotal [x]    = x binarioTotal (x:xs) = x*10^(length (x:xs)-1) + binarioTotal (xs)

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