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A286098 Square array read by antidiagonals: A(n,k) = T(n AND k, n OR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and OR is bitwise-or (A003986). 5
0, 1, 1, 3, 4, 3, 6, 6, 6, 6, 10, 11, 12, 11, 10, 15, 15, 17, 17, 15, 15, 21, 22, 21, 24, 21, 22, 21, 28, 28, 28, 28, 28, 28, 28, 28, 36, 37, 38, 37, 40, 37, 38, 37, 36, 45, 45, 47, 47, 49, 49, 47, 47, 45, 45, 55, 56, 55, 58, 59, 60, 59, 58, 55, 56, 55, 66, 66, 66, 66, 70, 70, 70, 70, 66, 66, 66, 66, 78, 79, 80, 79, 78, 83, 84, 83, 78, 79, 80, 79, 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

MathWorld, Pairing Function

FORMULA

A(n,k) = T(A004198(n,k), A003986(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,  4,   6,  11,  15,  22,  28,  37,  45,  56,  66,  79,  91

   3,  6,  12,  17,  21,  28,  38,  47,  55,  66,  80,  93, 105

   6, 11,  17,  24,  28,  37,  47,  58,  66,  79,  93, 108, 120

  10, 15,  21,  28,  40,  49,  59,  70,  78,  91, 105, 120, 140

  15, 22,  28,  37,  49,  60,  70,  83,  91, 106, 120, 137, 157

  21, 28,  38,  47,  59,  70,  84,  97, 105, 120, 138, 155, 175

  28, 37,  47,  58,  70,  83,  97, 112, 120, 137, 155, 174, 194

  36, 45,  55,  66,  78,  91, 105, 120, 144, 161, 179, 198, 218

  45, 56,  66,  79,  91, 106, 120, 137, 161, 180, 198, 219, 239

  55, 66,  80,  93, 105, 120, 138, 155, 179, 198, 220, 241, 261

  66, 79,  93, 108, 120, 137, 155, 174, 198, 219, 241, 264, 284

  78, 91, 105, 120, 140, 157, 175, 194, 218, 239, 261, 284, 312

MATHEMATICA

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

PROG

(Scheme)

(define (A286098 n) (A286098bi (A002262 n) (A025581 n)))

(define (A286098bi row col) (let ((a (A004198bi row col)) (b (A003986bi row col))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Here A003986bi and A004198bi implement bitwise-OR (A003986) 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, n|k)

for n in xrange(0, 21): print [A(k, n - k) for k in xrange(0, n + 1)] # Indranil Ghosh, May 21 2017

CROSSREFS

Cf. A000217 (row 0 & column 0), A084263 (seems to be row 1 & column 1), A046092 (main diagonal).

Cf. A003056, A003986, A004198.

Cf. also arrays A286099, A286101, A286102, A286108.

Sequence in context: A005092 A136195 A117892 * A074372 A049276 A101684

Adjacent sequences:  A286095 A286096 A286097 * A286099 A286100 A286101

KEYWORD

nonn,tabl

AUTHOR

Antti Karttunen, May 03 2017

STATUS

approved

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Last modified June 20 17:27 EDT 2019. Contains 324234 sequences. (Running on oeis4.)