A286153 Square array read by descending antidiagonals A(1,1), A(1,2), A(2,1), ...: If n > k, A(n,k) = T(n XOR k, k), and otherwise A(n,k) = T(n, n XOR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

2, 11, 13, 7, 5, 8, 22, 8, 7, 26, 16, 38, 9, 42, 19, 37, 47, 58, 62, 52, 43, 29, 23, 48, 14, 51, 25, 34, 56, 30, 39, 19, 16, 41, 33, 64, 46, 80, 31, 25, 20, 23, 32, 88, 53, 79, 93, 108, 32, 41, 39, 31, 116, 102, 89, 67, 57, 94, 140, 33, 27, 30, 148, 101, 63, 76, 106, 68, 81, 157, 176, 34, 29, 184, 166, 87, 75, 118, 92, 138, 69, 175, 158, 216, 35, 224, 165, 185, 74, 150, 103

Offset

1,1

Links

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array

Formula

A(n,k) = A286151(n,k), for n >= 1, k >= 1.

If n > k, A(n,k) = T(A003987(n,k),k), otherwise A(n,k) = T(n,A003987(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

Example

The top left 1 .. 12 x 1 .. 12 corner of the array:
    2,  11,   7,  22,  16,  37,  29,  56,  46,  79,  67, 106
   13,   5,   8,  38,  47,  23,  30,  80,  93,  57,  68, 138
    8,   7,   9,  58,  48,  39,  31, 108,  94,  81,  69, 174
   26,  42,  62,  14,  19,  25,  32, 140, 157, 175, 194,  82
   19,  52,  51,  16,  20,  41,  33, 176, 158, 215, 195, 110
   43,  25,  41,  23,  39,  27,  34, 216, 237, 177, 196, 142
   34,  33,  32,  31,  30,  29,  35, 260, 238, 217, 197, 178
   64,  88, 116, 148, 184, 224, 268,  44,  53,  63,  74,  86
   53, 102, 101, 166, 165, 246, 245,  46,  54,  87,  75, 114
   89,  63,  87, 185, 225, 183, 223,  57,  81,  65,  76, 146
   76,  75,  74, 205, 204, 203, 202,  69,  68,  67,  77, 182
  118, 150, 186,  86, 114, 146, 182,  82, 110, 142, 178,  90

Mathematica

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=If[n>k, T[BitXor[n, k], k], T[n, BitXor[n, k]]]; Table[A[k, n - k + 1], {n, 20}, {k, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

Prog

(Scheme) (define (A286153 n) (A286151bi (A002260 n) (A004736 n))) ;; For A286151bi see A286151.
(Python)
def T(a, b): return ((a + b)**2 + 3*a + b)/2
def A(n, k): return T(n^k, k) if n>k else T(n, n^k)
for n in range(1, 21): print [A(k, n - k + 1) for k in range(1, n + 1)] # Indranil Ghosh, May 21 2017

Crossrefs

Array A286151 without its topmost row and leftmost column.

Cf. A003987, A091255, A286155.

Sequence in context: A037091 A289675 A127303 * A068972 A116437 A048867

Adjacent sequences: A286150 A286151 A286152 * A286154 A286155 A286156

Keyword

nonn,tabl

Author

Antti Karttunen, May 03 2017

Status

approved