A285732 Square array A(n,k) read by antidiagonals, A(n,n) = -n, 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.

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

Offset: 1

Author

Antti Karttunen, May 03 2017

Keywords

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of the array
• MathWorld, Pairing Function

Crossrefs

Transpose: A285733.

Cf. A000124 (row 1, after -1), A000217 (column 1, after -1).

Cf. also A000027, A003989, A072030, A285721, A285722, A286100.

Programs

Python

def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
def A(n, k): return -n 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 (A285732 n) (A285732bi (A002260 n) (A004736 n)))
(define (A285732bi row col) (cond ((= row col) (- row)) ((> row col) (A000027bi (- row col) col)) (else (A000027bi row (- col row)))))

Formula

If n = k, A(n,k) = -n, 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.

A(n,k) = A285722(n,k) - A286100(n,k).