T(n,k)=Half the number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock having sum 4.

Terms

a(0) =31a(1) =216a(2) =216a(3) =1500a(4) =3552a(5) =1500a(6) =10428a(7) =57966a(8) =57966a(9) =10428a(10) =72468a(11) =948164a(12) =2214457a(13) =948164a(14) =72468a(15) =503676a(16) =15497634a(17) =84888699a(18) =84888699a(19) =15497634a(20) =503676a(21) =3500532a(22) =253372446a(23) =3250564943a(24) =7635071835a(25) =3250564943a(26) =253372446a(27) =3500532a(28) =24329052