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

1,4

Comments

See A285722.

Links

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

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

Transpose: A285722.

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

Cf. also A285733.

Sequence in context: A222602 A058544 A112156 * A072328 A135040 A048733

Adjacent sequences: A285720 A285721 A285722 * A285724 A285725 A285726

Keyword

nonn,tabl

Author

Antti Karttunen, May 03 2017

Status

approved