A231829 = Import["https://oeis.org/A231829/b231829.txt", "Table"][[All, 2]];
imax = Length[A231829];
mmax = Sqrt[2 imax] // Ceiling;
Clear[V](* requisite in case of re-run *); 
VV = Table[V[m - n + 1, n], {m, 1, mmax}, {n, 1, m}] // Flatten;
Do[Evaluate[VV[[i]]] = A231829[[i]], {i, 1, imax}];
V[0, _] = V[_, 0] = 0;
T[m_, n_] := If[m == 1 || n == 1, 1, U[m, n] - 2 U[m, n - 1] + U[m, n - 2]];
U[m_, n_] := V[m, n] - 2 V[m - 1, n] + V[m - 2, n];
Table[T[m - n + 1, n], {m, 1, mmax}, {n, 1, m}] // Flatten // Take[#, imax]&