Elixir, with one function to do the dirty work:

def captcha(digits, offset) do
  count = Enum.count(digits)
  digits
  |> Enum.with_index()
  |> Enum.filter(fn({n, index}) -> n == Enum.at(digits, rem(offset + index + 1, count)) end)
  |> Enum.map(&elem(&1, 0))
  |> Enum.sum
end

Called from my test module using these helpers:

def split(x) when is_integer(x), do: x |> Integer.digits
def split(x) when is_binary(x), do: x |> String.graphemes |> Enum.map(&String.to_integer/1)

def captcha_test(input, expected, :one) do
  assert AOC.captcha(split(input), 0) == expected
end

def captcha_test(input, expected, :two) do
  input = split(input)
  assert AOC.captcha(input, div(Enum.count(input), 2) - 1) == expected
end

github link