T(n,k)=Half the number of (n+1)X(n+1) symmetric 0..k arrays with no 2X2 subblock summing to 2k.

Terms

a(0) =3a(1) =11a(2) =13a(3) =28a(4) =192a(5) =70a(6) =56a(7) =1320a(8) =7827a(9) =529a(10) =99a(11) =5470a(12) =208072a(13) =750261a(14) =5148a(15) =159a(16) =17499a(17) =2322599a(18) =109820000a(19) =168382284a(20) =68798a(21) =240a(22) =45892a(23) =16537086a(24) =4286833981a(25) =193946752094a(26) =88681187619a(27) =1220409a(28) =344a(29) =105856