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 A082043 #20 Oct 05 2023 08:35:39
%S A082043 1,1,1,1,4,1,1,9,9,1,1,16,25,16,1,1,25,49,49,25,1,1,36,81,100,81,36,1,
%T A082043 1,49,121,169,169,121,49,1,1,64,169,256,289,256,169,64,1,1,81,225,361,
%U A082043 441,441,361,225,81,1,1,100,289,484,625,676,625,484,289,100,1
%N A082043 Square array, A(n, k) = (k*n)^2 + 2*k*n + 1, read by antidiagonals.
%H A082043 G. C. Greubel, <a href="/A082043/b082043.txt">Antidiagonals n = 0..50, flattened</a>
%F A082043 A(n, k) = (k*n)^2 + 2*k*n + 1 (square array).
%F A082043 T(n, k) = (k*(n-k))^2 + 2*k*(n-k) + 1 (number triangle).
%F A082043 A(k, n) = A(n, k).
%F A082043 T(n, n-k) = T(n, k).
%F A082043 A(n, n) = T(2*n, n) = A082044(n).
%F A082043 A(n, n-1) = T(2*n+1, n-1) = A058031(n), n >= 1.
%F A082043 A(n, n-2) = T(2*(n-1), n) = A000583(n-1), n >= 2.
%F A082043 A(n, n-3) = T(2*n-3, n) = A062938(n-3), n >= 3.
%F A082043 Sum_{k=0..n} T(n, k) = A082045(n) (diagonal sums).
%F A082043 Sum_{k=0..n} (-1)^k * T(n, k) = (1/4)*(1+(-1)^n)*(2 - 3*n). - _G. C. Greubel_, Dec 24 2022
%e A082043 Array, A(n, k), begins as:
%e A082043 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012;
%e A082043 1, 4, 9, 16, 25, 36, 49, 64, 81, ... A000290;
%e A082043 1, 9, 25, 49, 81, 121, 169, 225, 289, ... A016754;
%e A082043 1, 16, 49, 100, 169, 256, 361, 484, 625, ... A016778;
%e A082043 1, 25, 81, 169, 289, 441, 625, 841, 1089, ... A016814;
%e A082043 1, 36, 121, 256, 441, 676, 961, 1296, 1681, ... A016862;
%e A082043 1, 49, 169, 361, 625, 961, 1369, 1849, 2401, ... A016922;
%e A082043 1, 64, 225, 484, 841, 1296, 1849, 2500, 3249, ... A016994;
%e A082043 1, 81, 289, 625, 1089, 1681, 2401, 3249, 4225, ... A017078;
%e A082043 1, 100, 361, 784, 1369, 2116, 3025, 4096, 5329, ... A017174;
%e A082043 1, 121, 441, 961, 1681, 2601, 3721, 5041, 6561, ... A017282;
%e A082043 1, 144, 529, 1156, 2025, 3136, 4489, 6084, 7921, ... A017402;
%e A082043 1, 169, 625, 1369, 2401, 3721, 5329, 7225, 9409, ... A017534;
%e A082043 1, 196, 729, 1600, 2809, 4356, 6241, 8464, 11025, ... ;
%e A082043 Antidiagonals, T(n, k), begin as:
%e A082043 1;
%e A082043 1, 1;
%e A082043 1, 4, 1;
%e A082043 1, 9, 9, 1;
%e A082043 1, 16, 25, 16, 1;
%e A082043 1, 25, 49, 49, 25, 1;
%e A082043 1, 36, 81, 100, 81, 36, 1;
%e A082043 1, 49, 121, 169, 169, 121, 49, 1;
%e A082043 1, 64, 169, 256, 289, 256, 169, 64, 1;
%e A082043 1, 81, 225, 361, 441, 441, 361, 225, 81, 1;
%e A082043 1, 100, 289, 484, 625, 676, 625, 484, 289, 100, 1;
%t A082043 T[n_, k_]:= (k*(n-k))^2 +2*k*(n-k) +1;
%t A082043 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 24 2022 *)
%o A082043 (Magma)
%o A082043 A082043:= func< n,k | (k*(n-k))^2 +2*k*(n-k) +1 >;
%o A082043 [A082043(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Dec 24 2022
%o A082043 (SageMath)
%o A082043 def A082043(n,k): return (k*(n-k))^2 +2*k*(n-k) +1
%o A082043 flatten([[A082043(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Dec 24 2022
%Y A082043 Rows include A000290, A016754, A016778, A016814, A016862, A016922, A016994, A017078, A017174, A017282, A017402, A017534.
%Y A082043 Diagonals include A000583, A058031, A062938, A082044 (main diagonal).
%Y A082043 Diagonal sums (row sums if viewed as number triangle) are A082045.
%Y A082043 Cf. A082039, A082045, A082046, A082105.
%K A082043 easy,nonn,tabl
%O A082043 0,5
%A A082043 _Paul Barry_, Apr 03 2003