Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #13 Dec 09 2021 01:04:35
%S 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 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 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 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.
%C 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 -1, -1, 0, 1, 4, 9, 20, 41, 84, 169, 340, 681, 1364 ...: A084639;
%C -1, -1, 1, 3, 9, 19, 41, 83, 169, 339, 681, 1363, 2729;
%C -1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;
%C -1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;
%C B(n,k) = (n-1)*A001045(k) - T(n,k).
%C First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).
%H G. C. Greubel, <a href="/A173261/b173261.txt">Antidiagonals n = 0..50 of the array, flattened</a>
%F From _G. C. Greubel_, Dec 03 2021: (Start)
%F T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).
%F Sum_{k=0..n} T(n-k, k) = A024206(n).
%F 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 T(2*n-2, n) = A093178(n). (End)
%e The array T(n,k) starts in row n=2 with columns k>=0 as:
%e 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ... A000034;
%e 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3 ... A010684;
%e 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4 ... A010685;
%e 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5 ... A010686;
%e 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6 ... A010687;
%e 1, 7, 1, 7, 1, 7, 1, 7, 1, 7, 1, 7 ... A010688;
%e 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8 ... A010689;
%e 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9 ... A010690;
%e 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.
%e Antidiagonal triangle begins as:
%e 1;
%e 1, 2;
%e 1, 3, 1;
%e 1, 4, 1, 2;
%e 1, 5, 1, 3, 1;
%e 1, 6, 1, 4, 1, 2;
%e 1, 7, 1, 5, 1, 3, 1;
%e 1, 8, 1, 6, 1, 4, 1, 2;
%e 1, 9, 1, 7, 1, 5, 1, 3, 1;
%e 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;
%e 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1;
%e 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;
%e 1, 13, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1;
%e 1, 14, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2;
%t T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);
%t Table[T[n-k, k], {n,2,17}, {k,2,n}]//Flatten (* _G. C. Greubel_, Dec 03 2021 *)
%o (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 Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.
%Y Cf. A001045, A024206, A084639, A093178, A173114.
%K nonn,tabl,easy
%O 2,3
%A _Paul Curtz_, Feb 14 2010