Order 4k+2 Trimagic Series Impossibility

Theorem. There are no order 4k+2 trimagic series.
Proof.
For any entry x,
  x ≡ x3 (mod 2)
thus
  S1 ≡ S3 (mod 2)

For order N,
  S1 = N(N2+1)/2
  S3 = N3(N2+1)2/4

For order 4k+2,
  S1 = (4k+2)((4k+2)2+1)/2 = (2k+1)(16k2+16k+5) ≡ 1 (mod 2)
  S3 = (4k+2)3((4k+2)2+1)2/4 = (4k+2)(2k+1)(2k+1)(16k2+16k+5)2 ≡ 0 (mod 2)