Problem:
Score : 200 points

Problem Statement

You are given two integers K and S.

Three variable X, Y and Z takes integer values satisfying 0≤X,Y,Z≤K.

How many different assignments of values to X, Y and Z are there such that X + Y + Z = S?

Constraints

2≤K≤2500

0≤S≤3K

K and S are integers.

Input

The input is given from Standard Input in the following format:

K S

Output

Print the number of the triples of X, Y and Z that satisfy the condition.

Sample Input 1

2 2

Sample Output 1

6

There are six triples of X, Y and Z that satisfy the condition:

X = 0, Y = 0, Z = 2

X = 0, Y = 2, Z = 0

X = 2, Y = 0, Z = 0

X = 0, Y = 1, Z = 1

X = 1, Y = 0, Z = 1

X = 1, Y = 1, Z = 0

Sample Input 2

5 15

Sample Output 2

1

The maximum value of X + Y + Z is 15, achieved by one triple of X, Y and Z.

Solution in LuaJIT (2.0.4) (judge verdict: Accepted)

local mfl = math.floor
local TranFFT = {}

TranFFT.initialize = function(self)
  self.size = 18
  -- 2^18
  self.n = 262144
  -- 1007681537: prime,
  -- 1007681537 % 262144 = 1
  self.mod = 1007681537
  -- (6161^262144) % mod = 1
  self.w = 6161
  -- (1007677693 * 262144) % mod = 1
  self.ninv = 1007677693
  -- (534835031 * 6161) % mod = 1
  self.winv = 534835031
  self.p2 = {1}
  for i = 2, self.size do
    self.p2[i] = self.p2[i - 1] * 2
  end
  self.binv = {}
  for i = 1, self.n do
    local y, z = 0, i - 1
    for j = 1, self.size do
      y = y + (z % 2) * self.p2[self.size + 1 - j]
      z = mfl(z / 2)
    end
    self.binv[i] = y + 1
  end
  self.wmul = {1}
  for i = 2, self.n do
    self.wmul[i] = self:mul(self.wmul[i - 1], self.w)
  end
  self.winvmul = {1}
  for i = 2, self.n do
    self.winvmul[i] = self:mul(self.winvmul[i - 1], self.winv)
  end
end

-- (44893^2) % 1007681537 = 18375
-- 1007681537 = 22446 * 44893 + rem(13259)
-- x0, y0 <= 44892
-- x1, y1 <= 22446
-- max(x1y1*18375+(x1y0+x0y1)*44893+x0y0) < 10^14
TranFFT.mul = function(self, x, y)
  local x0, y0 = x % 44893, y % 44893
  local x1, y1 = mfl(x / 44893), mfl(y / 44893)
  return (x1 * y1 * 18375 + (x1 * y0 + x0 * y1) * 44893 + x0 * y0) % self.mod
end

TranFFT.add = function(self, x, y) return (x + y) % self.mod end

TranFFT.fft_common = function(self, ary, wmul)
  local ret = {}
  for i = 1, self.n do
    ret[i] = ary[self.binv[i]]
  end
  for i = 1, self.size do
    local step_size = self.p2[i]
    local step_count = self.p2[self.size + 1 - i]
    for istep = 1, step_count do
      local ofst = (istep - 1) * step_size * 2
      for j = 1, step_size do
        local a1, a2 = ret[ofst + j], ret[ofst + step_size + j]
        ret[ofst + j] = self:add(a1, self:mul(a2, wmul[1 + (j - 1) * step_count]))
        ret[ofst + step_size + j] = self:add(a1, self:mul(a2, wmul[1 + (j + step_size - 1) * step_count]))
      end
    end
  end
  return ret
end

TranFFT.fft = function(self, ary)
  return self:fft_common(ary, self.wmul)
end
TranFFT.ifft = function(self, ary)
  local ret = self:fft_common(ary, self.winvmul)
  for i = 1, self.n do
    ret[i] = self:mul(ret[i], self.ninv)
  end
  return ret
end

TranFFT:initialize()
local a = {}
for i = 1, TranFFT.n do
  a[i] = 0
end
local k, s = io.read("*n", "*n")
for i = 1, k + 1 do
  a[i] = 1
end

local at = TranFFT:fft(a)
for i = 1, TranFFT.n do
  local z = at[i]
  at[i] = TranFFT:mul(z, TranFFT:mul(z, z))
end
local b = TranFFT:ifft(at)
print(b[s + 1])