A286099 Square array read by antidiagonals: A(n,k) = T(n OR k, n AND 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).

0, 2, 2, 5, 4, 5, 9, 9, 9, 9, 14, 13, 12, 13, 14, 20, 20, 18, 18, 20, 20, 27, 26, 27, 24, 27, 26, 27, 35, 35, 35, 35, 35, 35, 35, 35, 44, 43, 42, 43, 40, 43, 42, 43, 44, 54, 54, 52, 52, 50, 50, 52, 52, 54, 54, 65, 64, 65, 62, 61, 60, 61, 62, 65, 64, 65, 77, 77, 77, 77, 73, 73, 73, 73, 77, 77, 77, 77, 90, 89, 88, 89, 90, 85, 84, 85, 90, 89, 88, 89, 90

Offset: 0

Author

Antti Karttunen, May 03 2017

Keywords

• nonn
• tabl

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), ...

Examples

			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,   4,   9,  13,  20,  26,  35,  43,  54,  64,  77,  89, 104
   5,   9,  12,  18,  27,  35,  42,  52,  65,  77,  88, 102, 119
   9,  13,  18,  24,  35,  43,  52,  62,  77,  89, 102, 116, 135
  14,  20,  27,  35,  40,  50,  61,  73,  90, 104, 119, 135, 148
  20,  26,  35,  43,  50,  60,  73,  85, 104, 118, 135, 151, 166
  27,  35,  42,  52,  61,  73,  84,  98, 119, 135, 150, 168, 185
  35,  43,  52,  62,  73,  85,  98, 112, 135, 151, 168, 186, 205
  44,  54,  65,  77,  90, 104, 119, 135, 144, 162, 181, 201, 222
  54,  64,  77,  89, 104, 118, 135, 151, 162, 180, 201, 221, 244
  65,  77,  88, 102, 119, 135, 150, 168, 181, 201, 220, 242, 267
  77,  89, 102, 116, 135, 151, 168, 186, 201, 221, 242, 264, 291
  90, 104, 119, 135, 148, 166, 185, 205, 222, 244, 267, 291, 312
		

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

Crossrefs

Cf. A000096 (row 0 & column 0), A162761 (seems to be row 1 & column 1), A046092 (main diagonal).

Cf. A003056, A003986, A004198.

Cf. also arrays A286098, A286101, A286102, A286109.

Programs

Mathematica

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

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

Scheme

(define (A286099 n) (A286099bi (A002262 n) (A025581 n)))
(define (A286099bi row col) (let ((a (A003986bi row col)) (b (A004198bi row col))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Here A003986bi and A004198bi implement bitwise-OR (A003986) and bitwise-AND (A004198).

Formula

A(n,k) = T(A003986(n,k), 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, ...].