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

0,4

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), ...

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

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, ...].

Example

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

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 *)

Prog

(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).
(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

Crossrefs

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

Cf. A003056, A003987, A004198.

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

Sequence in context: A100667 A329736 A324086 * A096438 A299418 A214456

Adjacent sequences: A286105 A286106 A286107 * A286109 A286110 A286111

Keyword

nonn,tabl

Author

Antti Karttunen, May 03 2017

Status

approved