A285723 Transpose of square array A285722.

0, 1, 1, 3, 0, 2, 6, 2, 3, 4, 10, 5, 0, 5, 7, 15, 9, 4, 6, 8, 11, 21, 14, 8, 0, 9, 12, 16, 28, 20, 13, 7, 10, 13, 17, 22, 36, 27, 19, 12, 0, 14, 18, 23, 29, 45, 35, 26, 18, 11, 15, 19, 24, 30, 37, 55, 44, 34, 25, 17, 0, 20, 25, 31, 38, 46, 66, 54, 43, 33, 24, 16, 21, 26, 32, 39, 47, 56, 78, 65, 53, 42, 32, 23, 0, 27, 33, 40, 48, 57, 67, 91, 77, 64, 52, 41, 31, 22, 28, 34, 41, 49, 58, 68, 79

Offset: 1

Author

Antti Karttunen, May 03 2017

Keywords

• nonn
• tabl

Comments

See A285722.

Examples

			The top left 14 X 14 corner of the array:
   0,  1,  3,  6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91
   1,  0,  2,  5,  9, 14, 20, 27, 35, 44, 54, 65, 77, 90
   2,  3,  0,  4,  8, 13, 19, 26, 34, 43, 53, 64, 76, 89
   4,  5,  6,  0,  7, 12, 18, 25, 33, 42, 52, 63, 75, 88
   7,  8,  9, 10,  0, 11, 17, 24, 32, 41, 51, 62, 74, 87
  11, 12, 13, 14, 15,  0, 16, 23, 31, 40, 50, 61, 73, 86
  16, 17, 18, 19, 20, 21,  0, 22, 30, 39, 49, 60, 72, 85
  22, 23, 24, 25, 26, 27, 28,  0, 29, 38, 48, 59, 71, 84
  29, 30, 31, 32, 33, 34, 35, 36,  0, 37, 47, 58, 70, 83
  37, 38, 39, 40, 41, 42, 43, 44, 45,  0, 46, 57, 69, 82
  46, 47, 48, 49, 50, 51, 52, 53, 54, 55,  0, 56, 68, 81
  56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66,  0, 67, 80
  67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78,  0, 79
  79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91,  0
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..7260; the first 120 antidiagonals of the array

Crossrefs

Transpose: A285722.

Cf. A000217 (row 1), A000124 (column 1, from 1 onward).

Cf. also A285733.

Programs

Mathematica

A[n_, n_] = 0;
A[n_, k_] /; k == n - 1 := (k^2 - k + 2)/2;
A[1, k_] := (k^2 - 3 k + 4)/2;
A[n_, k_] /; 1 <= k <= n - 2 := A[n, k] = A[n, k + 1] + 1;
A[n_, k_] /; k > n := A[n, k] = A[n - 1, k] + 1;
T[n_, k_] := A[k, n];
Table[T[n - k + 1, k], {n, 1, 14}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

Python

def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def A(n, k): return 0 if n == k else T(n - k, k) if n>k else T(n, k - n)
for n in range(1, 21): print([A(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, May 03 2017

Scheme

(define (A285723 n) (A285722bi (A004736 n) (A002260 n))) ;; For A285722bi see A285722.

Formula

A(n,k) = A285722(k,n).