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 A173261 #13 Dec 09 2021 01:04:35
%S A173261 1,1,2,1,3,1,1,4,1,2,1,5,1,3,1,1,6,1,4,1,2,1,7,1,5,1,3,1,1,8,1,6,1,4,
%T A173261 1,2,1,9,1,7,1,5,1,3,1,1,10,1,8,1,6,1,4,1,2,1,11,1,9,1,7,1,5,1,3,1,1,
%U A173261 12,1,10,1,8,1,6,1,4,1,2,1,13,1,11,1,9,1,7,1,5,1,3,1,1,14,1,12,1,10,1,8,1,6,1,4,1,2
%N A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.
%C A173261 One may define another array B(n,0) = -1, B(n,k) = T(n,k-1) + 2*B(n,k-1), n>=2, which also starts in columns k>=0, as follows:
%C A173261 -1, -1, 0, 1, 4, 9, 20, 41, 84, 169, 340, 681, 1364 ...: A084639;
%C A173261 -1, -1, 1, 3, 9, 19, 41, 83, 169, 339, 681, 1363, 2729;
%C A173261 -1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;
%C A173261 -1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;
%C A173261 B(n,k) = (n-1)*A001045(k) - T(n,k).
%C A173261 First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).
%H A173261 G. C. Greubel, <a href="/A173261/b173261.txt">Antidiagonals n = 0..50 of the array, flattened</a>
%F A173261 From _G. C. Greubel_, Dec 03 2021: (Start)
%F A173261 T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).
%F A173261 Sum_{k=0..n} T(n-k, k) = A024206(n).
%F A173261 Sum_{k=0..floor((n+2)/2)} T(n-2*k+2, k) = (1/16)*(2*n^2 4*n -5*(1 +(-1)^n) + 4*sin(n*Pi/2)) (diagonal sums).
%F A173261 T(2*n-2, n) = A093178(n). (End)
%e A173261 The array T(n,k) starts in row n=2 with columns k>=0 as:
%e A173261 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ... A000034;
%e A173261 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3 ... A010684;
%e A173261 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4 ... A010685;
%e A173261 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5 ... A010686;
%e A173261 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6 ... A010687;
%e A173261 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7 ... A010688;
%e A173261 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8 ... A010689;
%e A173261 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9 ... A010690;
%e A173261 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.
%e A173261 Antidiagonal triangle begins as:
%e A173261 1;
%e A173261 1, 2;
%e A173261 1, 3, 1;
%e A173261 1, 4, 1, 2;
%e A173261 1, 5, 1, 3, 1;
%e A173261 1, 6, 1, 4, 1, 2;
%e A173261 1, 7, 1, 5, 1, 3, 1;
%e A173261 1, 8, 1, 6, 1, 4, 1, 2;
%e A173261 1, 9, 1, 7, 1, 5, 1, 3, 1;
%e A173261 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;
%e A173261 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1;
%e A173261 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;
%e A173261 1, 13, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1;
%e A173261 1, 14, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;
%t A173261 T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);
%t A173261 Table[T[n-k, k], {n,2,17}, {k,2,n}]//Flatten (* _G. C. Greubel_, Dec 03 2021 *)
%o A173261 (Sage) flatten([[(1/2)*((n-k+3) - (n-k+1)*(-1)^k) for k in (2..n)] for n in (2..17)]) # _G. C. Greubel_, Dec 03 2021
%Y A173261 Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.
%Y A173261 Cf. A001045, A024206, A084639, A093178, A173114.
%K A173261 nonn,tabl,easy
%O A173261 2,3
%A A173261 _Paul Curtz_, Feb 14 2010