Chains of square numbers which remain squares when repeatedly adding digits. [1,16,169] appears to be the only chain with a length greater than 2.

AppendedSquareChains.hs

import Data.Maybe

isSquare :: Integer -> Bool
isSquare x
  = root ^ 2 == x
    where
      root = floor $ sqrt $ fromIntegral x

append :: [Integer] -> Integer -> [Integer]
append ys x
  = [x*10 + i | i <- ys]

squares :: [Integer] -> [Integer]
squares = filter isSquare

maybeHead :: [a] -> Maybe a
maybeHead xs
  | null xs   = Nothing
  | otherwise = Just (head xs)

nextSquare :: Integer -> Maybe Integer
nextSquare x
  = maybeHead $ squares $ append [1..9] x

squareChain :: Integer -> [Integer]
squareChain x = squareChain' [x]

squareChain' :: [Integer] -> [Integer]
squareChain' xs
  | isNothing nextSquareOfXs = xs
  | otherwise                = squareChain' (xs ++ [fromJust nextSquareOfXs])
    where
      nextSquareOfXs = nextSquare (last xs)

squareChains :: Integer -> [[Integer]]
squareChains n
  = map squareChain $ squares [1..n]

squareChainsOfLengthGreaterThan :: Integer -> [[Integer]]
squareChainsOfLengthGreaterThan n x
  = filter (\x -> length x >= x) $ squareChains n