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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). 7
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 (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

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 xrange(0, 21): print [A(k, n - k) for k in xrange(0, 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: A174608 A130327 A224361 * A239665 A178179 A284833

Adjacent sequences:  A286106 A286107 A286108 * A286110 A286111 A286112

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, May 03 2017

STATUS

approved

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Last modified May 26 22:16 EDT 2019. Contains 323597 sequences. (Running on oeis4.)