A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.

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, 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, 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

Offset: 2

Author

Paul Curtz, Feb 14 2010

Keywords

• nonn
• tabl
• easy

Comments

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:

-1, -1, 0, 1, 4, 9, 20, 41, 84, 169, 340, 681, 1364 ...: A084639;

-1, -1, 1, 3, 9, 19, 41, 83, 169, 339, 681, 1363, 2729;

-1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;

-1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;

B(n,k) = (n-1)*A001045(k) - T(n,k).

First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).

Examples

			The array T(n,k) starts in row n=2 with columns k>=0 as:
  1,  2, 1,  2, 1,  2, 1,  2, 1,  2, 1,  2 ... A000034;
  1,  3, 1,  3, 1,  3, 1,  3, 1,  3, 1,  3 ... A010684;
  1,  4, 1,  4, 1,  4, 1,  4, 1,  4, 1,  4 ... A010685;
  1,  5, 1,  5, 1,  5, 1,  5, 1,  5, 1,  5 ... A010686;
  1,  6, 1,  6, 1,  6, 1,  6, 1,  6, 1,  6 ... A010687;
  1,  7, 1,  7, 1,  7, 1,  7, 1,  7, 1,  7 ... A010688;
  1,  8, 1,  8, 1,  8, 1,  8, 1,  8, 1,  8 ... A010689;
  1,  9, 1,  9, 1,  9, 1,  9, 1,  9, 1,  9 ... A010690;
  1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.
Antidiagonal triangle begins as:
  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,  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, 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;
		

Links

• G. C. Greubel, Antidiagonals n = 0..50 of the array, flattened

Crossrefs

Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.

Cf. A001045, A024206, A084639, A093178, A173114.

Programs

Mathematica

T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);
Table[T[n-k, k], {n,2,17}, {k,2,n}]//Flatten (* G. C. Greubel, Dec 03 2021 *)

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

Formula

From G. C. Greubel, Dec 03 2021: (Start)

T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).

Sum_{k=0..n} T(n-k, k) = A024206(n).

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).

T(2*n-2, n) = A093178(n). (End)