Want to solve eqn

a =.; b =.; c =.; e =.;
Solve[3 + 4 a + 4 a^2 + 4 b + 4 b^2 + 4 c + 4 c^2 == 
  3 (1 + 4 e + 4 e^2), {a, b, c, e}, Integers]

output gives 2 conditions

{{e -> ConditionalExpression[-(1/2) - Sqrt[
     3 + 4 a + 4 a^2 + 4 b + 4 b^2 + 4 c + 4 c^2]/(
     2 Sqrt[3]), ((a | b | c | e) \[Element] Integers && 
       a >= 0) || ((a | b | c | e) \[Element] Integers && 
       a <= -1)]}, {e -> 
   ConditionalExpression[-(1/2) + Sqrt[
     3 + 4 a + 4 a^2 + 4 b + 4 b^2 + 4 c + 4 c^2]/(
     2 Sqrt[3]), ((a | b | c | e) \[Element] Integers && 
       a >= 0) || ((a | b | c | e) \[Element] Integers && a <= -1)]}}

when I consider the case where e is an odd integer

a =.; b =.; c =.; e =.;
FindInstance[
 e == -(1/2) + Sqrt[3 + 4 a + 4 a^2 + 4 b + 4 b^2 + 4 c + 4 c^2]/(
    2 Sqrt[3]) && Mod[e, 2] == 1 && a <= -1, {a, b, c, e}, Integers]

I get strange output

{{a -> {{-2, 0, 0, 0}} + {0, 1, 1, 1}, 
  b -> {{-2, 0, 0, 0}} + {0, 1, 1, 1}, 
  c -> {{-2, 0, 0, 0}} + {0, 1, 1, 1}, 
  e -> {{-2, 0, 0, 0}} + {0, 1, 1, 1}}}

What does this mean? How do you interpret the output? Any comments appreciated.