A286150 Square array read by antidiagonals: A(n,k) = T(n XOR k, min(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

0, 2, 2, 5, 1, 5, 9, 13, 13, 9, 14, 8, 3, 8, 14, 20, 26, 7, 7, 26, 20, 27, 19, 42, 6, 42, 19, 27, 35, 43, 52, 62, 62, 52, 43, 35, 44, 34, 25, 51, 10, 51, 25, 34, 44, 54, 64, 33, 41, 16, 16, 41, 33, 64, 54, 65, 53, 88, 32, 23, 15, 23, 32, 88, 53, 65, 77, 89, 102, 116, 31, 39, 39, 31, 116, 102, 89, 77, 90, 76, 63, 101, 148, 30, 21, 30, 148, 101, 63, 76, 90

Offset

0,2

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(A003987(n,k), min(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,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90
   2,   1,  13,   8,  26,  19,  43,  34,  64,  53,  89,  76, 118
   5,  13,   3,   7,  42,  52,  25,  33,  88, 102,  63,  75, 150
   9,   8,   7,   6,  62,  51,  41,  32, 116, 101,  87,  74, 186
  14,  26,  42,  62,  10,  16,  23,  31, 148, 166, 185, 205,  86
  20,  19,  52,  51,  16,  15,  39,  30, 184, 165, 225, 204, 114
  27,  43,  25,  41,  23,  39,  21,  29, 224, 246, 183, 203, 146
  35,  34,  33,  32,  31,  30,  29,  28, 268, 245, 223, 202, 182
  44,  64,  88, 116, 148, 184, 224, 268,  36,  46,  57,  69,  82
  54,  53, 102, 101, 166, 165, 246, 245,  46,  45,  81,  68, 110
  65,  89,  63,  87, 185, 225, 183, 223,  57,  81,  55,  67, 142
  77,  76,  75,  74, 205, 204, 203, 202,  69,  68,  67,  66, 178
  90, 118, 150, 186,  86, 114, 146, 182,  82, 110, 142, 178,  78

Mathematica

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitXor[n, k], Min[n,  k]]; Table[A[k, n - k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

Prog

(Scheme)
(define (A286150 n) (A286150bi (A002262 n) (A025581 n)))
(define (A286150bi row col) (let ((a (A003987bi row col)) (b (min col row))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Where A003987bi implements bitwise-xor (A003987).
(Python)
def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, min(n, k))
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017

Crossrefs

Cf. A000096 (row 0 & column 0), A000217 (main diagonal).

Cf. A003987, A001477, A286108, A286109, A286145, A286147, A286151.

Sequence in context: A079301 A079300 A128932 * A071950 A274847 A165922

Adjacent sequences: A286147 A286148 A286149 * A286151 A286152 A286153

Keyword

nonn,tabl

Author

Antti Karttunen, May 03 2017

Status

approved