A286155 Square array A(n,k) read by antidiagonals, A(n,n) = -n, otherwise, if n > k, A(n,k) = T(n XOR k,k), else A(n,k) = T(n,n XOR k), where T(n,k) is sequence A000027 considered as a two-dimensional table and XOR is bitwise-xor (A003987).

-1, 4, 6, 2, -2, 3, 11, 3, 2, 15, 7, 23, -3, 27, 10, 22, 30, 39, 43, 35, 28, 16, 12, 31, -4, 34, 14, 21, 37, 17, 24, 10, 7, 26, 20, 45, 29, 57, 18, 14, -5, 12, 19, 65, 36, 56, 68, 81, 19, 26, 24, 18, 89, 77, 66, 46, 38, 69, 109, 20, -6, 17, 117, 76, 44, 55, 79, 47, 58, 124, 141, 21, 16, 149, 133, 64, 54, 91, 67, 107, 48, 140, 125, 177, -7, 185, 132, 150, 53, 119

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 1 .. 12 x 1 .. 12 corner of the array:
  -1,   4,   2,  11,   7,  22,  16,  37,  29,  56,  46,  79
   6,  -2,   3,  23,  30,  12,  17,  57,  68,  38,  47, 107
   3,   2,  -3,  39,  31,  24,  18,  81,  69,  58,  48, 139
  15,  27,  43,  -4,  10,  14,  19, 109, 124, 140, 157,  59
  10,  35,  34,   7,  -5,  26,  20, 141, 125, 176, 158,  83
  28,  14,  26,  12,  24,  -6,  21, 177, 196, 142, 159, 111
  21,  20,  19,  18,  17,  16,  -7, 217, 197, 178, 160, 143
  45,  65,  89, 117, 149, 185, 225,  -8,  36,  44,  53,  63
  36,  77,  76, 133, 132, 205, 204,  29,  -9,  64,  54,  87
  66,  44,  64, 150, 186, 148, 184,  38,  58, -10,  55, 115
  55,  54,  53, 168, 167, 166, 165,  48,  47,  46, -11, 147
  91, 119, 151,  63,  87, 115, 147,  59,  83, 111, 143, -12
		

Links

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

Crossrefs

Cf. A003987, A091255.

Cf. also arrays A285732, A286151, A286153.

Programs

Scheme

(define (A286155 n) (A286155bi (A002260 n) (A004736 n)))
(define (A286155bi row col) (cond ((= row col) (- row)) ((> row col) (A000027bi (A003987bi row col) col)) (else (A000027bi row (A003987bi col row))))) ;; Where A003987bi implements bitwise-xor (A003987).
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

Formula

If n = k, A(n,k) = -n, if n > k, A(n,k) = T(A003987(n,k),k), otherwise [when n < k], A(n,k) = T(n,A003987(n,k)), 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.