A268719 Triangular table T(n>=0,k=0..n) = A003188(A006068(n) + A006068(k)), read by rows as A(0,0), A(1,0), A(1,1), A(2,0), A(2,1), A(2,2), ...

0, 1, 3, 2, 6, 5, 3, 2, 7, 6, 4, 12, 15, 13, 9, 5, 4, 13, 12, 11, 10, 6, 7, 4, 5, 14, 15, 12, 7, 5, 12, 4, 10, 14, 13, 15, 8, 24, 27, 25, 29, 31, 26, 30, 17, 9, 8, 25, 24, 31, 30, 27, 26, 19, 18, 10, 11, 8, 9, 26, 27, 24, 25, 22, 23, 20, 11, 9, 24, 8, 30, 26, 25, 27, 18, 22, 21, 23, 12, 13, 14, 15, 8, 9, 10, 11, 28, 29, 30, 31, 24

Offset

0,3

Links

Antti Karttunen, Table of n, a(n) for n = 0..15050; rows 0 .. 172 of the triangular table

Formula

T(n,k) = A003188(A006068(n) + A006068(k)).

a(n) = A268715(A003056(n), A002262(n)). [As a linear sequence.]

Example

The first fifteen rows of the triangle:
                             0
                           1   3
                         2   6   5
                       3   2   7   6
                     4  12  15  13   9
                   5   4  13  12  11  10
                 6   7   4   5  14  15  12
               7   5  12   4  10  14  13  15
             8  24  27  25  29  31  26  30  17
           9   8  25  24  31  30  27  26  19  18
        10  11   8   9  26  27  24  25  22  23  20
      11   9  24   8  30  26  25  27  18  22  21  23
    12  13  14  15   8   9  10  11  28  29  30  31  24
  13  15  10  14  24   8  11   9  20  28  31  29  25  27
14  10   9  11  27  25   8  24  23  21  28  20  26  30  29

Mathematica

a88[n_] := BitXor[n, Floor[n/2]];
a68[n_] := BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}];
a68[0] = 0;
T[n_, k_] := a88[a68[n] + a68[k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

Prog

(Scheme) (define (A268719 n) (A268715bi (A003056 n) (A002262 n)))
(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): a003188(a006068(n) + a006068(k))
for n in range(21): print [T(n, k) for k in range(n + 1)] # Indranil Ghosh, Jun 07 2017

Crossrefs

Cf. A003188, A006068.

Cf. A002262, A003056, A268715.

Cf. A001477 (left edge), A001969 (right edge).

Cf. A268720 (row sums).

Sequence in context: A304533 A090571 A088452 * A297878 A234922 A049777

Adjacent sequences: A268716 A268717 A268718 * A268720 A268721 A268722

Keyword

nonn,tabl

Author

Antti Karttunen, Feb 13 2016

Status

approved