A286109 Square array read by antidiagonals: A(n,k) = T(n XOR k, 2*(n AND 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, 2, 2, 5, 3, 5, 9, 9, 9, 9, 14, 12, 10, 12, 14, 20, 20, 16, 16, 20, 20, 27, 25, 27, 21, 27, 25, 27, 35, 35, 35, 35, 35, 35, 35, 35, 44, 42, 40, 42, 36, 42, 40, 42, 44, 54, 54, 50, 50, 46, 46, 50, 50, 54, 54, 65, 63, 65, 59, 57, 55, 57, 59, 65, 63, 65, 77, 77, 77, 77, 69, 69, 69, 69, 77, 77, 77, 77, 90, 88, 86, 88, 90, 80, 78, 80, 90, 88, 86, 88, 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), 2*A004198(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,   3,   9,  12,  20,  25,  35,  42,  54,  63,  77,  88, 104
   5,   9,  10,  16,  27,  35,  40,  50,  65,  77,  86, 100, 119
   9,  12,  16,  21,  35,  42,  50,  59,  77,  88, 100, 113, 135
  14,  20,  27,  35,  36,  46,  57,  69,  90, 104, 119, 135, 144
  20,  25,  35,  42,  46,  55,  69,  80, 104, 117, 135, 150, 162
  27,  35,  40,  50,  57,  69,  78,  92, 119, 135, 148, 166, 181
  35,  42,  50,  59,  69,  80,  92, 105, 135, 150, 166, 183, 201
  44,  54,  65,  77,  90, 104, 119, 135, 136, 154, 173, 193, 214
  54,  63,  77,  88, 104, 117, 135, 150, 154, 171, 193, 212, 236
  65,  77,  86, 100, 119, 135, 148, 166, 173, 193, 210, 232, 259
  77,  88, 100, 113, 135, 150, 166, 183, 193, 212, 232, 253, 283
  90, 104, 119, 135, 144, 162, 181, 201, 214, 236, 259, 283, 300

Mathematica

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

Prog

(Scheme)
(define (A286109 n) (A286109bi (A002262 n) (A025581 n)))
(define (A286109bi row col) (let ((a (A003987bi row col)) (b (* 2 (A004198bi 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(n^k, 2*(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. A000096 (row 0 & column 0), A014105 (main diagonal).

Cf. A003056, A003987, A004198.

Cf. also arrays A286099, A286108, A286145, A286147, A286150, A286151.

Sequence in context: A224361 A341443 A224166 * A239665 A178179 A284833

Adjacent sequences: A286106 A286107 A286108 * A286110 A286111 A286112

Keyword

nonn,tabl

Author

Antti Karttunen, May 03 2017

Status

approved