A286108 Square array read by antidiagonals: A(n,k) = T(2*(n AND k), n XOR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and XOR is bitwise-xor (A003987).

0, 1, 1, 3, 5, 3, 6, 6, 6, 6, 10, 12, 14, 12, 10, 15, 15, 19, 19, 15, 15, 21, 23, 21, 27, 21, 23, 21, 28, 28, 28, 28, 28, 28, 28, 28, 36, 38, 40, 38, 44, 38, 40, 38, 36, 45, 45, 49, 49, 53, 53, 49, 49, 45, 45, 55, 57, 55, 61, 63, 65, 63, 61, 55, 57, 55, 66, 66, 66, 66, 74, 74, 74, 74, 66, 66, 66, 66, 78, 80, 82, 80, 78, 88, 90, 88, 78, 80, 82, 80, 78

Offset: 0

Author

Antti Karttunen, May 03 2017

Keywords

• nonn
• tabl

Comments

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Examples

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,  1,   3,   6,  10,  15,  21,  28,  36,  45,  55,  66,  78
   1,  5,   6,  12,  15,  23,  28,  38,  45,  57,  66,  80,  91
   3,  6,  14,  19,  21,  28,  40,  49,  55,  66,  82,  95, 105
   6, 12,  19,  27,  28,  38,  49,  61,  66,  80,  95, 111, 120
  10, 15,  21,  28,  44,  53,  63,  74,  78,  91, 105, 120, 144
  15, 23,  28,  38,  53,  65,  74,  88,  91, 107, 120, 138, 161
  21, 28,  40,  49,  63,  74,  90, 103, 105, 120, 140, 157, 179
  28, 38,  49,  61,  74,  88, 103, 119, 120, 138, 157, 177, 198
  36, 45,  55,  66,  78,  91, 105, 120, 152, 169, 187, 206, 226
  45, 57,  66,  80,  91, 107, 120, 138, 169, 189, 206, 228, 247
  55, 66,  82,  95, 105, 120, 140, 157, 187, 206, 230, 251, 269
  66, 80,  95, 111, 120, 138, 157, 177, 206, 228, 251, 275, 292
  78, 91, 105, 120, 144, 161, 179, 198, 226, 247, 269, 292, 324
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
• Eric Weisstein's World of Mathematics, Pairing Function

Crossrefs

Cf. A000217 (row 0 & column 0), A014106 (main diagonal).

Cf. A003056, A003987, A004198.

Cf. also arrays A286098, A286109, A286145, A286147, A286150, A286151.

Programs

Mathematica

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[2*BitAnd[n, k], BitXor[n, k]]; Table[A[k, n - k ], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 20 2017 *)

Python

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(2*(n&k), n^k)
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 20 2017

Scheme

(define (A286108 n) (A286108bi (A002262 n) (A025581 n)))
(define (A286108bi row col) (let ((a (* 2 (A004198bi row col))) (b (A003987bi row col))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Here A003987bi and A004198bi implement bitwise-xor (A003987) and bitwise-and (A004198).

Formula

A(n,k) = T(2*A004198(n,k), A003987(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].