I'm working on strengthening my F#-fu and decided to tackle the Facebook Hacker Cup Double Squares problem.  I'm having some problems with the run-time and was wondering if anyone could help me figure out why it is so much slower than my C# equivalent.
There's a good description from another post;
  Source: Facebook Hacker Cup
  Qualification Round 2011
  
  A double-square number is an integer X
  which can be expressed as the sum of
  two perfect squares. For example, 10
  is a double-square because 10 = 3^2 +
  1^2. Given X, how can we determine the number of ways in which it can be
  written as the sum of two squares? For
  example, 10 can only be written as 3^2
  + 1^2 (we don't count 1^2 + 3^2 as being different). On the other hand, 25 can
  be written as 5^2 + 0^2 or as 4^2 + 3^2.
  
  You need to solve this problem for 0 =
  X = 2,147,483,647.
  
  Examples:
  
  10 = 1 
  
  25 = 2 
  
  3 = 0 
  
  0 = 1 
  
  1 = 1
My basic strategy (which I'm open to critique on) is to; 
Create a dictionary (for memoize) of the input numbers initialzed to 0
Get the largest number (LN) and pass it to count/memo function
Get the LN square root as int
Calculate squares for all numbers 0 to LN and store in dict  
Sum squares for non repeat combinations of numbers from 0 to LN
If sum is in memo dict, add 1 to memo  
Finally, output the counts of the original numbers.
Here is the F# code (See code changes at bottom) I've written that I believe corresponds to this strategy (Runtime: ~8:10);
open System
open System.Collections.Generic
open System.IO
/// Get a sequence of values
let rec range min max = 
    seq { for num in [min .. max] do yield num }
/// Get a sequence starting from 0 and going to max
let rec zeroRange max = range 0 max    
/// Find the maximum number in a list with a starting accumulator (acc)
let rec maxNum acc = function
    | [] -> acc
    | p::tail when p > acc -> maxNum p tail
    | p::tail -> maxNum acc tail
/// A helper for finding max that sets the accumulator to 0
let rec findMax nums = maxNum 0 nums
/// Build a collection of combinations; ie [1,2,3] = (1,1), (1,2), (1,3), (2,2), (2,3), (3,3)
let rec combos range =    
    seq { 
          let count = ref 0
          for inner in range do 
              for outer in Seq.skip !count range do 
                  yield (inner, outer)
              count := !count + 1          
        }
let rec squares nums = 
    let dict = new Dictionary<int, int>()
    for s in nums do
        dict.[s] <- (s * s)
    dict
/// Counts the number of possible double squares for a given number and keeps track of other counts that are provided in the memo dict.
let rec countDoubleSquares (num: int) (memo: Dictionary<int, int>) =
    // The highest relevent square is the square root because it squared plus 0 squared is the top most possibility
    let maxSquare = System.Math.Sqrt((float)num)
    // Our relevant squares are 0 to the highest possible square; note the cast to int which shouldn't hurt.
    let relSquares = range 0 ((int)maxSquare)
    // calculate the squares up front;
    let calcSquares = squares relSquares
    // Build up our square combinations; ie [1,2,3] = (1,1), (1,2), (1,3), (2,2), (2,3), (3,3)
    for (sq1, sq2) in combos relSquares do
        let v = calcSquares.[sq1] + calcSquares.[sq2]
        // Memoize our relevant results
        if memo.ContainsKey(v) then            
            memo.[v] <- memo.[v] + 1
    // return our count for the num passed in
    memo.[num]
// Read our numbers from file.
//let lines = File.ReadAllLines("test2.txt")
//let nums = [ for line in Seq.skip 1 lines -> Int32.Parse(line) ]
// Optionally, read them from straight array
let nums = [1740798996; 1257431873; 2147483643; 602519112; 858320077; 1048039120; 415485223; 874566596; 1022907856; 65; 421330820; 1041493518; 5; 1328649093; 1941554117; 4225; 2082925; 0; 1; 3]
// Initialize our memoize dictionary
let memo = new Dictionary<int, int>()
for num in nums do
    memo.[num] <- 0
// Get the largest number in our set, all other numbers will be memoized along the way
let maxN = findMax nums
// Do the memoize
let maxCount = countDoubleSquares maxN memo
// Output our results.
for num in nums do
    printfn "%i" memo.[num]
// Have a little pause for when we debug
let line = Console.Read()
And here is my version in C# (Runtime: ~1:40:
using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Text;
namespace FBHack_DoubleSquares
{
    public class TestInput
    {
        public int NumCases { get; set; }
        public List<int> Nums { get; set; }
        public TestInput()
        {
            Nums = new List<int>();
        }
        public int MaxNum()
        {
            return Nums.Max();
        }
    }
    class Program
    {
        static void Main(string[] args)
        {
            // Read input from file.
            //TestInput input = ReadTestInput("live.txt");
            // As example, load straight.
            TestInput input = new TestInput
            {
                NumCases = 20,
                Nums = new List<int>
                {
                    1740798996,
                    1257431873,
                    2147483643,
                    602519112,
                    858320077,
                    1048039120,
                    415485223,
                    874566596,
                    1022907856,
                    65,
                    421330820,
                    1041493518,
                    5,
                    1328649093,
                    1941554117,
                    4225,
                    2082925,
                    0,
                    1,
                    3,
                }
            };
            var maxNum = input.MaxNum();
            Dictionary<int, int> memo = new Dictionary<int, int>();
            foreach (var num in input.Nums)
            {
                if (!memo.ContainsKey(num))
                    memo.Add(num, 0);
            }
            DoMemoize(maxNum, memo);
            StringBuilder sb = new StringBuilder();
            foreach (var num in input.Nums)
            {
                //Console.WriteLine(memo[num]);
                sb.AppendLine(memo[num].ToString());
            }
            Console.Write(sb.ToString());
            var blah = Console.Read();
            //File.WriteAllText("out.txt", sb.ToString());
        }
        private static int DoMemoize(int num, Dictionary<int, int> memo)
        {
            var highSquare = (int)Math.Floor(Math.Sqrt(num));
            var squares = CreateSquareLookup(highSquare);
            var relSquares = squares.Keys.ToList();
            Debug.WriteLine("Starting - " + num.ToString());
            Debug.WriteLine("RelSquares.Count = {0}", relSquares.Count);
            int sum = 0;
            var index = 0;            
            foreach (var square in relSquares)
            {
                foreach (var inner in relSquares.Skip(index))
                {
                    sum = squares[square] + squares[inner];
                    if (memo.ContainsKey(sum))
                        memo[sum]++;
                }
                index++;
            }
            if (memo.ContainsKey(num))
                return memo[num];
            return 0;            
        }
        private static TestInput ReadTestInput(string fileName)
        {
            var lines = File.ReadAllLines(fileName);
            var input = new TestInput();
            input.NumCases = int.Parse(lines[0]);
            foreach (var lin in lines.Skip(1))
            {
                input.Nums.Add(int.Parse(lin));
            }
            return input;
        }
        public static Dictionary<int, int> CreateSquareLookup(int maxNum)
        {
            var dict = new Dictionary<int, int>();
            int square;
            foreach (var num in Enumerable.Range(0, maxNum))
            {
                square = num * num;
                dict[num] = square;
            }
            return dict;
        }
    }   
}
Thanks for taking a look.
UPDATE
Changing the combos function slightly will result in a pretty big performance boost (from 8 min to 3:45):
/// Old and Busted...
let rec combosOld range =    
    seq { 
          let rangeCache = Seq.cache range
          let count = ref 0
          for inner in rangeCache do 
              for outer in Seq.skip !count rangeCache do 
                  yield (inner, outer)
              count := !count + 1          
        }
/// The New Hotness...
let rec combos maxNum =    
    seq {
        for i in 0..maxNum do
            for j in i..maxNum do
                yield i,j }