A108644 Square array A(n,k) read by ascending antidiagonals: A(n,n) = n^2, if n>k: A(n,k) = n*(n-1) + k, if k>n: A(n,k) = n + (k-1)^2.

1, 3, 2, 7, 4, 5, 13, 8, 6, 10, 21, 14, 9, 11, 17, 31, 22, 15, 12, 18, 26, 43, 32, 23, 16, 19, 27, 37, 57, 44, 33, 24, 20, 28, 38, 50, 73, 58, 45, 34, 25, 29, 39, 51, 65, 91, 74, 59, 46, 35, 30, 40, 52, 66, 82, 111, 92, 75, 60, 47, 36, 41, 53, 67, 83, 101

Offset: 1

Author

Pierre CAMI, Jun 27 2005

Keywords

• nonn
• tabl

Comments

The table gives all positive integers exactly once.

Examples

			Array begins:
   1  2  5 10 17 26 37 ...
   3  4  6 11 18 27 38 ...
   7  8  9 12 19 28 39 ...
  13 14 15 16 20 29 40 ...
  21 22 23 24 25 30 41 ...
  31 32 33 34 35 36 42 ...
  43 44 45 46 47 48 49 ...
  ...
Antidiagonal triangle begins as:
   1;
   3,  2;
   7,  4,  5;
  13,  8,  6, 10;
  21, 14,  9, 11, 17;
  31, 22, 15, 12, 18, 26;
  43, 32, 23, 16, 19, 27, 37;
  ...
		

Links

• G. C. Greubel, Antidiagonals n = 1..50, flattened
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A002522 (1st row), A002061 (1st column), A000290 (diagonal).

Cf. A002378, A002522, A005563, A033954, A098557, A274248.

Programs

Magma

A:= func< n,k | k lt n select k+n*(n-1) else k eq n select n^2 else n+(k-1)^2 >;
A108644:= func< n,k | A(n-k+1,k) >;
[A108644(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 18 2023

Mathematica

A[n_, k_]:= If[kA108644[n_, k_]:= A[n-k+1,k];
Table[A108644[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 18 2023 *)

PARI

A(i,j)=if (i==j, i^2, if (i>j, i*(i-1)+j, (j-1)^2+i));
matrix(7,7,n,k,A(n,k)) \\ Michel Marcus, Dec 30 2020

SageMath

def A(n,k):
    if kA108644(n,k): return A(n-k+1,k)
flatten([[A108644(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 18 2023

Formula

From G. C. Greubel, Oct 18 2023: (Start)

T(n, k) = A(n-k+1, k) (antidiagonal triangle).

T(n, n) = A002522(n-1).

T(2*n, n) = A005563(n).

T(2*n-1, n) = A000290(n).

T(2*n-2, n) = A002378(n-1), n >= 2.

T(3*n, n) = A033954(n).

Sum_{k=1..n} T(n, k) = A274248(n). (End)

Let M be the upper left n X n submatrix of this array, then abs(det(M)) = A098557(n). - Thomas Scheuerle, Nov 11 2023