triangular.rb

# Babylonian method
def isqrt(n)
  return 0 if n == 0
  s, t = 1, n
  while s < t
    s *= 2
    t /= 2
  end
  begin
    t = s
    s = (n / s + s) / 2
  end while s < t
  t
end

def triangular(n)
  n * (n + 1) / 2
end

# x (x + 1) / 2 <= n を変形すると
# (2x + 1)^2 <= 8n + 1
def triangular_inv(n)
  a = isqrt(8 * n + 1)
  if a % 2 == 0
    (a - 2) / 2
  else
    (a - 1) / 2
  end
end

def istriangular(n)
  triangular(triangular_inv(n)) == n
end

def issquare(n)
  isqrt(n)**2 == n
end

1000.times do |i|
  [1, -1].each do |a|
    n = 2 ** i + a
    if istriangular(n)
        puts "#{n} is triangular"
    end
    if issquare(n)
        puts "#{n} is square"
    end
  end
end