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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). 7
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 (list; table; graph; refs; listen; history; text; internal format)
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 xrange(0, 21): print [A(k, n - k) for k in xrange(0, 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: A151568 A134429 A100667 * A096438 A299418 A214456

Adjacent sequences:  A286105 A286106 A286107 * A286109 A286110 A286111

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, May 03 2017

STATUS

approved

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Last modified May 19 06:57 EDT 2019. Contains 323386 sequences. (Running on oeis4.)