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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
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
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 range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 20 2017
CROSSREFS
Cf. A000217 (row 0 & column 0), A014106 (main diagonal).
Sequence in context: A100667 A329736 A324086 * A096438 A299418 A214456
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 03 2017
STATUS
approved