A285722 Square array A(n,k) read by antidiagonals, A(n,n) = 0, otherwise, if n > k, A(n,k) = T(n-k,k), else A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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

Offset: 1

Author

Antti Karttunen, May 03 2017

Keywords

• nonn
• tabl

Comments

The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Examples

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10585
• MathWorld, Pairing Function

Crossrefs

Transpose: A285723.

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

Cf. also A000027, A003989, A072030, A285721, A285732.

Programs

Mathematica

A[n_, n_] = 0;
A[n_, k_] /; k == n-1 := (k^2 - k + 2)/2;
A[1, k_] := (k^2 - 3k + 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;
Table[A[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(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 03 2017

Scheme

(define (A285722 n) (A285722bi (A002260 n) (A004736 n)))
(define (A285722bi row col) (cond ((= row col) 0) ((> row col) (A000027bi (- row col) col)) (else (A000027bi row (- col row)))))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

Formula

If n = k, A(n,k) = 0, if n > k, A(n,k) = T(n-k,k), otherwise [when n < k], A(n,k) = T(n,k-n), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N X N to N.