Sumas parciales de Juzuk

PorJosé A. Alonso 23-enero-201830-enero-2018

En 1939 Dov Juzuk extendió el método de Nicómaco del cálculo de los cubos. La extensión se basaba en los siguientes pasos:

se comienza con la lista de todos los enteros positivos

     [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

1	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...

se agrupan tomando el primer elemento, los dos siguientes, los tres
siguientes, etc.

     [[1], [2, 3], [4, 5, 6], [7, 8, 9, 10], [11, 12, 13, 14, 15], ...

1	[[1], [2, 3], [4, 5, 6], [7, 8, 9, 10], [11, 12, 13, 14, 15], ...

se seleccionan los elementos en posiciones pares

     [[1],         [4, 5, 6],                [11, 12, 13, 14, 15], ...

1	[[1], [4, 5, 6], [11, 12, 13, 14, 15], ...

se suman los elementos de cada grupo

     [1,           15,                       65,                   ...

1	[1, 15, 65, ...

se calculan las sumas acumuladas

     [1,           16,                       81,                   ...

1	[1, 16, 81, ...

Las sumas obtenidas son las cuantas potencias de los números enteros positivos.

Definir las funciones

   listasParcialesJuzuk :: [a] -> [[a]]
   sumasParcialesJuzuk  :: [Integer] -> [Integer]

1 2	listasParcialesJuzuk :: [a] -> [[a]] sumasParcialesJuzuk :: [Integer] -> [Integer]

tal que

(listasParcialesJuzuk xs) es lalista de ls listas parciales de Juzuk; es decir, la selección de los elementos en posiciones pares de la agrupación de los elementos de xs tomando el primer elemento, los dos siguientes, los tres siguientes, etc. Por ejemplo,

     λ> take 4 (listasParcialesJuzuk [1..])
     [[1],[4,5,6],[11,12,13,14,15],[22,23,24,25,26,27,28]]
     λ> take 4 (listasParcialesJuzuk [1,3..])
     [[1],[7,9,11],[21,23,25,27,29],[43,45,47,49,51,53,55]]

λ> take 4 (listasParcialesJuzuk [1..])

[[1],[4,5,6],[11,12,13,14,15],[22,23,24,25,26,27,28]]

λ> take 4 (listasParcialesJuzuk [1,3..])

[[1],[7,9,11],[21,23,25,27,29],[43,45,47,49,51,53,55]]

(sumasParcialesJuzuk xs) es la lista de las sumas acumuladas de los elementos de las listas de Juzuk generadas por xs. Por ejemplo,

     take 4 (sumasParcialesJuzuk [1..])  ==  [1,16,81,256]
     take 4 (sumasParcialesJuzuk [1,3..])  ==  [1,28,153,496]

1 2	take 4 (sumasParcialesJuzuk [1..]) == [1,16,81,256] take 4 (sumasParcialesJuzuk [1,3..]) == [1,28,153,496]

Comprobar con QuickChek que, para todo entero positivo n,

el elemento de (sumasParcialesJuzuk [1..]) en la posición (n-1) es n^4.
el elemento de (sumasParcialesJuzuk [1,3..]) en la posición (n-1) es n^2*(2*n^2 - 1).
el elemento de (sumasParcialesJuzuk [1,5..]) en la posición (n-1) es 4*n^4-3*n^2.
el elemento de (sumasParcialesJuzuk [2,3..]) en la posición (n-1) es n^2*(n^2+1).

Soluciones

import Data.List (genericIndex)
import Test.QuickCheck

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk = elementosEnPares . listasParciales

-- (listasParciales xs) es la agrupación de los elementos de xs obtenida
-- tomando el primer elemento, los dos siguientes, los tres siguientes,
-- etc. Por ejemplo, 
--    λ> take 5 (listasParciales [1..])
--    [[1],[2,3],[4,5,6],[7,8,9,10],[11,12,13,14,15]]
listasParciales :: [a] -> [[a]]
listasParciales = aux 1
  where aux n xs = ys : aux (n+1) zs  
          where (ys,zs) = splitAt n xs

-- (elementosEnPares xs) es la lista de los elementos de xs en
-- posiciones pares. Por ejemplo,
--    λ> elementosEnPares [[1],[2,3],[4,5,6],[7,8,9,10],[11,12,13,14,15]]
--    [[1],[4,5,6],[11,12,13,14,15]]
elementosEnPares :: [a] -> [a]
elementosEnPares []       = []
elementosEnPares [x]      = [x]
elementosEnPares (x:_:xs) = x : elementosEnPares xs

sumasParcialesJuzuk :: [Integer] -> [Integer]
sumasParcialesJuzuk xs =
  scanl1 (+) (map sum (listasParcialesJuzuk xs))

-- La primera propiedad es
prop_sumasParcialesJuzuk :: (Positive Integer) -> Bool
prop_sumasParcialesJuzuk (Positive n) =
  sumasParcialesJuzuk [1..] `genericIndex` (n-1) == n^4

-- Su comprobación es
--    λ> quickCheck prop_sumasParcialesJuzuk
--    +++ OK, passed 100 tests.

-- La segunda propiedad es
prop_sumasParcialesJuzuk2 :: (Positive Integer) -> Bool
prop_sumasParcialesJuzuk2 (Positive n) =
  sumasParcialesJuzuk [1,3..] `genericIndex` (n-1) == n^2*(2*n^2 - 1)

-- Su comprobación es
--    λ> quickCheck prop_sumasParcialesJuzuk2
--    +++ OK, passed 100 tests.

-- La tercera propiedad es
prop_sumasParcialesJuzuk3 :: (Positive Integer) -> Bool
prop_sumasParcialesJuzuk3 (Positive n) =
  sumasParcialesJuzuk [1,5..] `genericIndex` (n-1) == 4*n^4-3*n^2

-- Su comprobación es
--    λ> quickCheck prop_sumasParcialesJuzuk3
--    +++ OK, passed 100 tests.

-- La cuarta propiedad es
prop_sumasParcialesJuzuk4 :: (Positive Integer) -> Bool
prop_sumasParcialesJuzuk4 (Positive n) =
  sumasParcialesJuzuk [2,3..] `genericIndex` (n-1) == n^2*(n^2+1)
  
-- Su comprobación es
--    λ> quickCheck prop_sumasParcialesJuzuk4
--    +++ OK, passed 100 tests.

import Data.List (genericIndex)

import Test.QuickCheck

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk = elementosEnPares . listasParciales

-- (listasParciales xs) es la agrupación de los elementos de xs obtenida

-- tomando el primer elemento, los dos siguientes, los tres siguientes,

-- etc. Por ejemplo,

-- λ> take 5 (listasParciales [1..])

-- [[1],[2,3],[4,5,6],[7,8,9,10],[11,12,13,14,15]]

listasParciales :: [a] -> [[a]]

listasParciales = aux 1

where aux n xs = ys : aux (n+1) zs

where (ys,zs) = splitAt n xs

-- (elementosEnPares xs) es la lista de los elementos de xs en

-- posiciones pares. Por ejemplo,

-- λ> elementosEnPares [[1],[2,3],[4,5,6],[7,8,9,10],[11,12,13,14,15]]

-- [[1],[4,5,6],[11,12,13,14,15]]

elementosEnPares :: [a] -> [a]

elementosEnPares [] = []

elementosEnPares [x] = [x]

elementosEnPares (x:_:xs) = x : elementosEnPares xs

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk xs =

scanl1 (+) (map sum (listasParcialesJuzuk xs))

-- La primera propiedad es

prop_sumasParcialesJuzuk :: (Positive Integer) -> Bool

prop_sumasParcialesJuzuk (Positive n) =

sumasParcialesJuzuk [1..] `genericIndex` (n-1) == n^4

-- Su comprobación es

-- λ> quickCheck prop_sumasParcialesJuzuk

-- +++ OK, passed 100 tests.

-- La segunda propiedad es

prop_sumasParcialesJuzuk2 :: (Positive Integer) -> Bool

prop_sumasParcialesJuzuk2 (Positive n) =

sumasParcialesJuzuk [1,3..] `genericIndex` (n-1) == n^2*(2*n^2 - 1)

-- Su comprobación es

-- λ> quickCheck prop_sumasParcialesJuzuk2

-- +++ OK, passed 100 tests.

-- La tercera propiedad es

prop_sumasParcialesJuzuk3 :: (Positive Integer) -> Bool

prop_sumasParcialesJuzuk3 (Positive n) =

sumasParcialesJuzuk [1,5..] `genericIndex` (n-1) == 4*n^4-3*n^2

-- Su comprobación es

-- λ> quickCheck prop_sumasParcialesJuzuk3

-- +++ OK, passed 100 tests.

-- La cuarta propiedad es

prop_sumasParcialesJuzuk4 :: (Positive Integer) -> Bool

prop_sumasParcialesJuzuk4 (Positive n) =

sumasParcialesJuzuk [2,3..] `genericIndex` (n-1) == n^2*(n^2+1)

-- Su comprobación es

-- λ> quickCheck prop_sumasParcialesJuzuk4

-- +++ OK, passed 100 tests.

11 Comentarios

import Test.QuickCheck
listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk xs = [ys | ys <- listasParciales xs, odd (length ys), not(null ys)]

listasParciales :: [a] -> [[a]]
listasParciales = aux 0
  where aux n xs = ys : aux (n+1) zs  
          where (ys,zs) = splitAt n xs

sumasParcialesJuzuk :: [Integer] -> [Integer]
sumasParcialesJuzuk xs = aux (listasParcialesJuzuk xs)
  where aux (xs:xss) = sum xs : map (+sum xs) (aux xss)

propiedad1 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1..])!!(n-1)) ==  n^4
--quickCheck propiedad1   
-- +++ OK, passed 100 tests.

propiedad2 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1,3..])!!(n-1)) ==  (n^2)*(2*n^2 -1)
--quickCheck propiedad2  
-- +++ OK, passed 100 tests.

propiedad3 n = n>0 ==> fromInteger((sumasParcialesJuzuk [1,5..]) !! (n-1)) == 4*n^4-3*n^2
--quickCheck propiedad3  
-- +++ OK, passed 100 tests.


propiedad4 n = n>0 ==> fromInteger((sumasParcialesJuzuk [2,3..]) !! (n-1)) == n^2*(n^2+ 1)
-- quickCheck propiedad4  
-- +++ OK, passed 100 tests.

import Test.QuickCheck

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk xs = [ys | ys <- listasParciales xs, odd (length ys), not(null ys)]

listasParciales :: [a] -> [[a]]

listasParciales = aux 0

where aux n xs = ys : aux (n+1) zs

where (ys,zs) = splitAt n xs

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk xs = aux (listasParcialesJuzuk xs)

where aux (xs:xss) = sum xs : map (+sum xs) (aux xss)

propiedad1 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1..])!!(n-1)) == n^4

--quickCheck propiedad1

-- +++ OK, passed 100 tests.

propiedad2 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1,3..])!!(n-1)) == (n^2)*(2*n^2 -1)

--quickCheck propiedad2

-- +++ OK, passed 100 tests.

propiedad3 n = n>0 ==> fromInteger((sumasParcialesJuzuk [1,5..]) !! (n-1)) == 4*n^4-3*n^2

--quickCheck propiedad3

-- +++ OK, passed 100 tests.

propiedad4 n = n>0 ==> fromInteger((sumasParcialesJuzuk [2,3..]) !! (n-1)) == n^2*(n^2+ 1)

-- quickCheck propiedad4

-- +++ OK, passed 100 tests.

Responder

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk xs = [xss !! n | n <- [0,2..]]
  where xss = [take n (drop (div (n*(n-1)) 2) xs) | n <- [1..]]

sumasParcialesJuzuk  :: [Integer] -> [Integer]
sumasParcialesJuzuk xs  = [sum(take n xss) | n <- [1..]]
  where xss = map sum (listasParcialesJuzuk xs)

prop1 :: Int -> Property
prop1 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1..]) !! (n-1)) ==  n^4
prop2 :: Int -> Property
prop2 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1,3..]) !! (n-1)) ==  (n^2)*(2*n^2 -1)
prop3 :: Int -> Property
prop3 n = n>0 ==> fromInteger((sumasParcialesJuzuk [1,5..]) !! (n-1)) == 4*n^4-3*n^2
prop4 :: Int -> Property
prop4 n = n>0 ==> fromInteger((sumasParcialesJuzuk [2,3..]) !! (n-1)) == n^2*(n^2+ 1)

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk xs = [xss !! n | n <- [0,2..]]

where xss = [take n (drop (div (n*(n-1)) 2) xs) | n <- [1..]]

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk xs = [sum(take n xss) | n <- [1..]]

where xss = map sum (listasParcialesJuzuk xs)

prop1 :: Int -> Property

prop1 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1..]) !! (n-1)) == n^4

prop2 :: Int -> Property

prop2 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1,3..]) !! (n-1)) == (n^2)*(2*n^2 -1)

prop3 :: Int -> Property

prop3 n = n>0 ==> fromInteger((sumasParcialesJuzuk [1,5..]) !! (n-1)) == 4*n^4-3*n^2

prop4 :: Int -> Property

prop4 n = n>0 ==> fromInteger((sumasParcialesJuzuk [2,3..]) !! (n-1)) == n^2*(n^2+ 1)

Responder

import Data.List       (scanl')
import Data.List.Split (splitPlaces)
import Test.QuickCheck

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk xs =
  [splitPlaces [1..] xs !! n | n <- [0,2..]]

sumasParcialesJuzuk :: [Integer] -> [Integer]
sumasParcialesJuzuk =
  tail
  . scanl' (flip ((+) . sum)) 0
  . listasParcialesJuzuk

prop_Juzuk1 :: (Positive Integer) -> Bool
prop_Juzuk1 (Positive n) =
  sumasParcialesJuzuk [1..]
  !! (fromEnum n - 1) == n^4

prop_Juzuk2 :: (Positive Integer) -> Bool
prop_Juzuk2 (Positive n) =
  sumasParcialesJuzuk [1,3..]
  !! (fromEnum n - 1) == n^2 * (2 * n^2 - 1)

prop_Juzuk3 :: (Positive Integer) -> Bool
prop_Juzuk3 (Positive n) =
  sumasParcialesJuzuk [1,5..]
  !! (fromEnum n - 1) == 4 * n^4 - 3 * n^2

prop_Juzuk4 :: (Positive Integer) -> Bool
prop_Juzuk4 (Positive n) =
  sumasParcialesJuzuk [2,3..]
  !! (fromEnum n - 1) == n^2 * (n^2 + 1)

import Data.List (scanl')

import Data.List.Split (splitPlaces)

import Test.QuickCheck

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk xs =

[splitPlaces [1..] xs !! n | n <- [0,2..]]

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk =

tail

. scanl' (flip ((+) . sum)) 0

. listasParcialesJuzuk

prop_Juzuk1 :: (Positive Integer) -> Bool

prop_Juzuk1 (Positive n) =

sumasParcialesJuzuk [1..]

!! (fromEnum n - 1) == n^4

prop_Juzuk2 :: (Positive Integer) -> Bool

prop_Juzuk2 (Positive n) =

sumasParcialesJuzuk [1,3..]

!! (fromEnum n - 1) == n^2 * (2 * n^2 - 1)

prop_Juzuk3 :: (Positive Integer) -> Bool

prop_Juzuk3 (Positive n) =

sumasParcialesJuzuk [1,5..]

!! (fromEnum n - 1) == 4 * n^4 - 3 * n^2

prop_Juzuk4 :: (Positive Integer) -> Bool

prop_Juzuk4 (Positive n) =

sumasParcialesJuzuk [2,3..]

!! (fromEnum n - 1) == n^2 * (n^2 + 1)

Responder

Para comprobar las propiedades:

{-# LANGUAGE TemplateHaskell #-}

return []
runTests = $quickCheckAll

-- La comprobación es
--    λ> runTests
--    === prop_Juzuk1 ===
--    +++ OK, passed 100 tests.
--
--    === prop_Juzuk2 ===
--    +++ OK, passed 100 tests.
--
--    === prop_Juzuk3 ===
--    +++ OK, passed 100 tests.
--
--    === prop_Juzuk4 ===
--    +++ OK, passed 100 tests.
--
--    True

{-# LANGUAGE TemplateHaskell #-}

return []

runTests = $quickCheckAll

-- La comprobación es

-- λ> runTests

-- === prop_Juzuk1 ===

-- +++ OK, passed 100 tests.

-- === prop_Juzuk2 ===

-- +++ OK, passed 100 tests.

-- === prop_Juzuk3 ===

-- +++ OK, passed 100 tests.

-- === prop_Juzuk4 ===

-- +++ OK, passed 100 tests.

-- True

Responder

import Test.QuickCheck
import Data.List
import Data.List.Split

listasParciales :: [a] -> [[a]]
listasParciales xs = splitPlaces [1..] xs

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk xs = [x | (x,i) <- zip (listasParciales xs) [1..], i `mod` 2 /= 0]

sumas :: [Integer] -> [Integer]
sumas xs = map (sum) (listasParcialesJuzuk xs)

sumasParcialesJuzuk :: [Integer] -> [Integer]
sumasParcialesJuzuk xs = scanl1 (+) (sumas xs)

propiedad_1 :: Int -> Property
propiedad_1 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1..]) !! (n-1)) ==  n^4

propiedad_2 :: Int -> Property
propiedad_2 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1,3..]) !! (n-1)) ==  (n^2)*(2*n^2 -1)

propiedad_3 :: Int -> Property
propiedad_3 n = n>0 ==> fromInteger((sumasParcialesJuzuk [1,5..]) !! (n-1)) == 4*n^4-3*n^2

propiedad_4 :: Int -> Property
propiedad_4 n = n>0 ==> fromInteger((sumasParcialesJuzuk [2,3..]) !! (n-1)) == n^2*(n^2+ 1)

import Test.QuickCheck

import Data.List

import Data.List.Split

listasParciales :: [a] -> [[a]]

listasParciales xs = splitPlaces [1..] xs

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk xs = [x | (x,i) <- zip (listasParciales xs) [1..], i `mod` 2 /= 0]

sumas :: [Integer] -> [Integer]

sumas xs = map (sum) (listasParcialesJuzuk xs)

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk xs = scanl1 (+) (sumas xs)

propiedad_1 :: Int -> Property

propiedad_1 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1..]) !! (n-1)) == n^4

propiedad_2 :: Int -> Property

propiedad_2 n = n>0 ==>fromInteger((sumasParcialesJuzuk [1,3..]) !! (n-1)) == (n^2)*(2*n^2 -1)

propiedad_3 :: Int -> Property

propiedad_3 n = n>0 ==> fromInteger((sumasParcialesJuzuk [1,5..]) !! (n-1)) == 4*n^4-3*n^2

propiedad_4 :: Int -> Property

propiedad_4 n = n>0 ==> fromInteger((sumasParcialesJuzuk [2,3..]) !! (n-1)) == n^2*(n^2+ 1)

Responder

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk = aux 1
  where aux _ [] = []
        aux y xs = take y xs : (aux (y+2) (drop (2*y+1) xs))


sumasParcialesJuzuk  :: [Integer] -> [Integer]
sumasParcialesJuzuk  = aux 0.listasParcialesJuzuk
                where aux _ [] = []
                      aux y (xs:xss) = z : aux z xss
                       where z = y + sum xs

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk = aux 1

where aux _ [] = []

aux y xs = take y xs : (aux (y+2) (drop (2*y+1) xs))

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk = aux 0.listasParcialesJuzuk

where aux _ [] = []

aux y (xs:xss) = z : aux z xss

where z = y + sum xs

Responder

import Data.List.Split

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk xs = aux (splitPlaces [1..] xs)
  where aux (x:y:xs) = x :aux xs

sumasParcialesJuzuk :: [Integer] -> [Integer]
sumasParcialesJuzuk = scanl1 (+) . map sum . listasParcialesJuzuk

import Data.List.Split

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk xs = aux (splitPlaces [1..] xs)

where aux (x:y:xs) = x :aux xs

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk = scanl1 (+) . map sum . listasParcialesJuzuk

Responder

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk = aux2 . aux 1 
  where aux n xs = ys : aux (n+1) zs
          where (ys,zs) = splitAt n xs

aux2 (xs:ys:xss) = xs: aux2 xss
                
sumasParcialesJuzuk  :: [Integer] -> [Integer]
sumasParcialesJuzuk = aux3 . listasParcialesJuzuk
  where aux3 (xs:ys:xss) = sum xs : aux3 ((sum xs:ys):xss)

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk = aux2 . aux 1

where aux n xs = ys : aux (n+1) zs

where (ys,zs) = splitAt n xs

aux2 (xs:ys:xss) = xs: aux2 xss

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk = aux3 . listasParcialesJuzuk

where aux3 (xs:ys:xss) = sum xs : aux3 ((sum xs:ys):xss)

Las comprobaciones serían análogas al primer comentario

Responder

import           Data.List
import           Test.QuickCheck

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk xs = aux xs [1..]
  where
    aux xs (k:p:ks) = take k xs : aux (drop (k+p) xs) ks

sumasParcialesJuzuk  :: [Integer] -> [Integer]
sumasParcialesJuzuk = scanl1 (+) . map sum . listasParcialesJuzuk

import Data.List

import Test.QuickCheck

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk xs = aux xs [1..]

where

aux xs (k:p:ks) = take k xs : aux (drop (k+p) xs) ks

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk = scanl1 (+) . map sum . listasParcialesJuzuk

Responder

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk xs = [ys | (ys,x) <- zip (agrupa1 1 xs) [0..], even x] 

agrupa1 :: Int -> [a] -> [[a]]
agrupa1 n (x:xs) = ys : agrupa1 (n+1) (drop (length ys - 1) xs )
  where ys = take n (x:xs)

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk xs = [ys | (ys,x) <- zip (agrupa1 1 xs) [0..], even x]

agrupa1 :: Int -> [a] -> [[a]]

agrupa1 n (x:xs) = ys : agrupa1 (n+1) (drop (length ys - 1) xs )

where ys = take n (x:xs)

Responder

listasParcialesJuzuk :: [a] -> [[a]]
listasParcialesJuzuk = aux 1
  where aux _ [] = []
        aux x xs = take x xs : (aux (x+2) (drop (2*x+1) xs))


sumasParcialesJuzuk  :: [Integer] -> [Integer]
sumasParcialesJuzuk xs  = [ sum (take k  ns) | k <- [1..]]
  where ns = map (sum) (listasParcialesJuzuk xs)

listasParcialesJuzuk :: [a] -> [[a]]

listasParcialesJuzuk = aux 1

where aux _ [] = []

aux x xs = take x xs : (aux (x+2) (drop (2*x+1) xs))

sumasParcialesJuzuk :: [Integer] -> [Integer]

sumasParcialesJuzuk xs = [ sum (take k ns) | k <- [1..]]

where ns = map (sum) (listasParcialesJuzuk xs)

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Sumas parciales de Juzuk

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