A286145 - OEIS

A286145

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

0, 4, 2, 12, 1, 5, 24, 18, 13, 9, 40, 17, 3, 8, 14, 60, 50, 11, 7, 26, 20, 84, 49, 61, 6, 42, 19, 27, 112, 98, 85, 73, 62, 52, 43, 35, 144, 97, 59, 72, 10, 51, 25, 34, 44, 180, 162, 83, 71, 22, 16, 41, 33, 64, 54, 220, 161, 181, 70, 38, 15, 23, 32, 88, 53, 65, 264, 242, 221, 201, 58, 48, 39, 31, 116, 102, 89, 77, 312, 241, 179, 200, 222, 47, 21, 30, 148, 101, 63, 76, 90

(list; table; graph; refs; listen; history; text; internal format)

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

MathWorld, Pairing Function

FORMULA

A(n,k) = T(A003987(n,k), 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, 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312

2, 1, 18, 17, 50, 49, 98, 97, 162, 161, 242, 241, 338

5, 13, 3, 11, 61, 85, 59, 83, 181, 221, 179, 219, 365

9, 8, 7, 6, 73, 72, 71, 70, 201, 200, 199, 198, 393

14, 26, 42, 62, 10, 22, 38, 58, 222, 266, 314, 366, 218

20, 19, 52, 51, 16, 15, 48, 47, 244, 243, 340, 339, 240

27, 43, 25, 41, 23, 39, 21, 37, 267, 315, 265, 313, 263

35, 34, 33, 32, 31, 30, 29, 28, 291, 290, 289, 288, 287

44, 64, 88, 116, 148, 184, 224, 268, 36, 56, 80, 108, 140

54, 53, 102, 101, 166, 165, 246, 245, 46, 45, 94, 93, 158

65, 89, 63, 87, 185, 225, 183, 223, 57, 81, 55, 79, 177

77, 76, 75, 74, 205, 204, 203, 202, 69, 68, 67, 66, 197

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], k]; Table[A[k, n - k ], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

PROG

(Scheme)

(define (A286145 n) (A286145bi (A002262 n) (A025581 n)))

(define (A286145bi row col) (let ((a (A003987bi row col)) (b col)) (/ (+ (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, k)

for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017

CROSSREFS

Transpose: A286147.

Cf. A046092 (row 0), A000096 (column 0), A000217 (main diagonal).

Cf. A003987, A001477, A286108, A286109, A286150, A286151.

Sequence in context: A142706 A361216 A092952 * A010318 A188134 A226725

Adjacent sequences: A286142 A286143 A286144 * A286146 A286147 A286148

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, May 03 2017

STATUS

approved