A268719 - OEIS

%I #19 Dec 07 2019 12:18:28

%S 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,

%T 5,12,4,10,14,13,15,8,24,27,25,29,31,26,30,17,9,8,25,24,31,30,27,26,

%U 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

%N 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), ...

%H Antti Karttunen, <a href="/A268719/b268719.txt">Table of n, a(n) for n = 0..15050; rows 0 .. 172 of the triangular table</a>

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

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

%e The first fifteen rows of the triangle:

%e                              0

%e                            1   3

%e                          2   6   5

%e                        3   2   7   6

%e                      4  12  15  13   9

%e                    5   4  13  12  11  10

%e                  6   7   4   5  14  15  12

%e                7   5  12   4  10  14  13  15

%e              8  24  27  25  29  31  26  30  17

%e            9   8  25  24  31  30  27  26  19  18

%e         10  11   8   9  26  27  24  25  22  23  20

%e       11   9  24   8  30  26  25  27  18  22  21  23

%e     12  13  14  15   8   9  10  11  28  29  30  31  24

%e   13  15  10  14  24   8  11   9  20  28  31  29  25  27

%e 14  10   9  11  27  25   8  24  23  21  28  20  26  30  29

%t a88[n_] := BitXor[n, Floor[n/2]];

%t a68[n_] := BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}];

%t a68[0] = 0;

%t T[n_, k_] := a88[a68[n] + a68[k]];

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 19 2019 *)

%o (Scheme) (define (A268719 n) (A268715bi (A003056 n) (A002262 n)))

%o (Python)

%o def a003188(n): return n^(n>>1)

%o def a006068(n):

%o     s=1

%o     while True:

%o         ns=n>>s

%o         if ns==0: break

%o         n=n^ns

%o         s<<=1

%o     return n

%o def T(n, k): a003188(a006068(n) + a006068(k))

%o for n in range(21): print [T(n, k) for k in range(n + 1)] # _Indranil Ghosh_, Jun 07 2017

%Y Cf. A003188, A006068.

%Y Cf. A002262, A003056, A268715.

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

%Y Cf. A268720 (row sums).

%K nonn,tabl

%O 0,3

%A _Antti Karttunen_, Feb 13 2016