A268715 Square array A(i,j) = A003188(A006068(i) + A006068(j)), read by antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

0, 1, 1, 2, 3, 2, 3, 6, 6, 3, 4, 2, 5, 2, 4, 5, 12, 7, 7, 12, 5, 6, 4, 15, 6, 15, 4, 6, 7, 7, 13, 13, 13, 13, 7, 7, 8, 5, 4, 12, 9, 12, 4, 5, 8, 9, 24, 12, 5, 11, 11, 5, 12, 24, 9, 10, 8, 27, 4, 14, 10, 14, 4, 27, 8, 10, 11, 11, 25, 25, 10, 15, 15, 10, 25, 25, 11, 11, 12, 9, 8, 24, 29, 14, 12, 14, 29, 24, 8, 9, 12, 13, 13, 24, 9, 31, 31, 13, 13, 31, 31, 9, 24, 13, 13

Offset

0,4

Comments

Each row n is row A006068(n) of array A268820 without its A006068(n) initial terms.

Links

Antti Karttunen, Table of n, a(n) for n = 0..15050; the first 173 antidiagonals of the array

Formula

A(i,j) = A003188(A006068(i) + A006068(j)) = A003188(A268714(i,j)).

A(row,col) = A268820(A006068(row), (A006068(row)+col)).

Example

The top left [0 .. 15] x [0 .. 15] section of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15
   1,  3,  6,  2, 12,  4,  7,  5, 24,  8, 11,  9, 13, 15, 10, 14
   2,  6,  5,  7, 15, 13,  4, 12, 27, 25,  8, 24, 14, 10,  9, 11
   3,  2,  7,  6, 13, 12,  5,  4, 25, 24,  9,  8, 15, 14, 11, 10
   4, 12, 15, 13,  9, 11, 14, 10, 29, 31, 26, 30,  8, 24, 27, 25
   5,  4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26,  9,  8, 25, 24
   6,  7,  4,  5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11,  8,  9
   7,  5, 12,  4, 10, 14, 13, 15, 30, 26, 25, 27, 11,  9, 24,  8
   8, 24, 27, 25, 29, 31, 26, 30, 17, 19, 22, 18, 28, 20, 23, 21
   9,  8, 25, 24, 31, 30, 27, 26, 19, 18, 23, 22, 29, 28, 21, 20
  10, 11,  8,  9, 26, 27, 24, 25, 22, 23, 20, 21, 30, 31, 28, 29
  11,  9, 24,  8, 30, 26, 25, 27, 18, 22, 21, 23, 31, 29, 20, 28
  12, 13, 14, 15,  8,  9, 10, 11, 28, 29, 30, 31, 24, 25, 26, 27
  13, 15, 10, 14, 24,  8, 11,  9, 20, 28, 31, 29, 25, 27, 30, 26
  14, 10,  9, 11, 27, 25,  8, 24, 23, 21, 28, 20, 26, 30, 29, 31
  15, 14, 11, 10, 25, 24,  9,  8, 21, 20, 29, 28, 27, 26, 31, 30

Mathematica

A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := BitXor @@ Table[Floor[ n/2^m], {m, 0, Log[2, n]}]; A006068[0]=0; A[i_, j_] := A003188[A006068[i] + A006068[j]]; Table[A[i-j, j], {i, 0, 13}, {j, 0, i}] // Flatten (* Jean-François Alcover, Feb 17 2016 *)

Prog

(Scheme)
(define (A268715 n) (A268715bi (A002262 n) (A025581 n)))
(define (A268715bi row col) (A003188 (+ (A006068 row) (A006068 col))))
;; Alternatively, extracting data from array A268820:
(define (A268715bi row col) (A268820bi (A006068 row) (+ (A006068 row) col)))
(Python)
def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def T(n, k): return a003188(a006068(n) + a006068(k))
for n in range(21): print([T(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017

Crossrefs

Cf. A003188, A006068, A268714, A268820.

Main diagonal: A001969.

Row 0, column 0: A001477.

Row 1, column 1: A268717.

Antidiagonal sums: A268837.

Cf. A268719 (the lower triangular section).

Cf. also A268725.

Sequence in context: A085208 A332553 A257302 * A085211 A085212 A079025

Adjacent sequences: A268712 A268713 A268714 * A268716 A268717 A268718

Keyword

nonn,tabl

Author

Antti Karttunen, Feb 12 2016

Status

approved