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 A347533 #19 Dec 25 2022 08:31:06
%S A347533 1,2,3,3,7,6,4,13,18,10,5,21,36,31,15,6,31,60,64,50,21,7,43,90,109,
%T A347533 105,71,28,8,57,126,166,180,151,98,36,9,73,168,235,275,261,210,127,45,
%U A347533 10,91,216,316,390,401,364,274,162,55,11,111,270,409,525,571,560,477,351,199,66
%N A347533 Array A(n,k) where A(n,0) = n and A(n,k) = (k*n + 1)^2 - A(n,k-1), n > 0, read by ascending antidiagonals.
%C A347533 A(n,k) is also the distance from A(n, k-1) to the earliest square greater than 3*A(n,k-1) - A(n,k-2).
%C A347533 In column k, every term is the arithmetic mean of its neighbors minus A000217(k).
%H A347533 G. C. Greubel, <a href="/A347533/b347533.txt">Antidiagonals n = 1..50, flattened</a>
%F A347533 A(n,k) = A000217(k)*n^2 + k*n + 1, for k odd.
%F A347533 A(n,k) = A000217(k)*n^2 + (k+1)*n = (k+1)*x*(k*n/2 + 1), for k even.
%F A347533 A(n,k) = (A(n,k-1) + A(n,k+1) + k*(k+1))/2, for any k.
%F A347533 A(n, 0) = A000027(n).
%F A347533 A(n, 1) = A002061(n+1).
%F A347533 A(n, 2) = A028896(n).
%F A347533 A(n, 3) = A085473(n).
%F A347533 From _G. C. Greubel_, Dec 25 2022: (Start)
%F A347533 A(n, k) = (1/2)*( (k*n+1)*(k*n+n+1) + (-1)^k*(n-1) ).
%F A347533 T(n, k) = (1/2)*( (k*(n-k)+1)*((k+1)*(n-k)+1) + (-1)^k*(n-k-1) ).
%F A347533 Sum_{k=0..n-1} T(n, k) = (1/120)*(2*n^5 + 5*n^4 + 20*n^3 + 25*n^2 + 98*n - 15*(1-(-1)^n)). (End)
%e A347533 Array, A(n, k), begins:
%e A347533 1 3 6 10 15 21 28 36 45 ... A000217;
%e A347533 2 7 18 31 50 71 98 127 162 ... A195605;
%e A347533 3 13 36 64 105 151 210 274 351 ...
%e A347533 4 21 60 109 180 261 364 477 612 ...
%e A347533 5 31 90 166 275 401 560 736 945 ...
%e A347533 6 43 126 235 390 571 798 1051 1350 ...
%e A347533 7 57 168 316 525 771 1078 1422 1827 ...
%e A347533 8 73 216 409 680 1001 1400 1849 2376 ...
%e A347533 9 91 270 514 855 1261 1764 2332 2997 ...
%e A347533 Antidiagonals, T(n, k), begin as:
%e A347533 1;
%e A347533 2, 3;
%e A347533 3, 7, 6;
%e A347533 4, 13, 18, 10;
%e A347533 5, 21, 36, 31, 15;
%e A347533 6, 31, 60, 64, 50, 21;
%e A347533 7, 43, 90, 109, 105, 71, 28;
%e A347533 8, 57, 126, 166, 180, 151, 98, 36;
%e A347533 9, 73, 168, 235, 275, 261, 210, 127, 45;
%e A347533 10, 91, 216, 316, 390, 401, 364, 274, 162, 55;
%t A347533 A[n_, 0]:= n; A[n_, k_]:= (k*n+1)^2 -A[n,k-1]; Table[Function[n, A[n, k]][m-k+1], {m,0,10}, {k,0,m}]//Flatten (* _Michael De Vlieger_, Oct 27 2021 *)
%o A347533 (Magma)
%o A347533 A347533:= func< n,k | (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1)) >;
%o A347533 [A347533(n,k): k in [0..n-1], n in [1..13]]; // _G. C. Greubel_, Dec 25 2022
%o A347533 (SageMath)
%o A347533 def A347533(n,k): return (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1))
%o A347533 flatten([[A347533(n,k) for k in range(n)] for n in range(1,14)]) # _G. C. Greubel_, Dec 25 2022
%Y A347533 Family of sequences (k*n + 1)^2: A016754 (k=2), A016778 (k=3), A016814 (k=4), A016862 (k=5), A016922 (k=6), A016994 (k=7), A017078 (k=8), A017174 (k=9), A017282 (k=10), A017402 (k=11), A017534 (k=12), A134934 (k=14).
%Y A347533 Cf. A000027, A000217, A002061, A028896, A085473, A195605.
%K A347533 nonn,tabl,easy
%O A347533 1,2
%A A347533 _Lamine Ngom_, Sep 05 2021