This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383742 #22 May 08 2025 08:56:00
%S A383742 0,0,1,0,1,2,0,1,2,3,0,1,6,5,4,0,1,14,35,12,5,0,1,34,197,204,29,6,0,1,
%T A383742 82,1155,2772,1189,70,7,0,1,198,6725,39236,39005,6930,169,8,0,1,478,
%U A383742 39203,551532,1332869,548842,40391,408,9,0,1,1154,228485,7761996,45232349,45278310,7722793,235416,985,10
%N A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)*x + (-1)^k*x^2).
%F A383742 A(0,k) = 0, A(1,k) = 1; A(n,k) = A002203(k) * A(n-1,k) - (-1)^k * A(n-2,k) for n > 1.
%F A383742 A(n,k) = Pell(k*n)/Pell(k) for k > 0.
%e A383742 Square array begins:
%e A383742 0, 0, 0, 0, 0, 0, ...
%e A383742 1, 1, 1, 1, 1, 1, ...
%e A383742 2, 2, 6, 14, 34, 82, ...
%e A383742 3, 5, 35, 197, 1155, 6725, ...
%e A383742 4, 12, 204, 2772, 39236, 551532, ...
%e A383742 5, 29, 1189, 39005, 1332869, 45232349, ...
%t A383742 A[n_, k_] := Fibonacci[k*n, 2]/Fibonacci[k, 2]; A[n_, 0] := n; Table[A[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 08 2025 *)
%o A383742 (PARI) pell(n) = ([2, 1; 1, 0]^n)[2, 1];
%o A383742 a(n, k) = if(k==0, n, pell(k*n)/pell(k));
%Y A383742 Columns k=0..6 give A001477, A000129, A001109, A041085(n-1), A091761, A292423, A097731(n-1).
%Y A383742 Rows n=0..5 give A000004, A000012, A002203, A383720, A383740, A383741.
%Y A383742 Main diagonal gives A380083.
%Y A383742 Cf. A028412.
%K A383742 nonn,tabl,easy
%O A383742 0,6
%A A383742 _Seiichi Manyama_, May 07 2025