cp's OEIS Frontend

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), ...

%I A268719 #23 Jun 24 2025 10:45:40
%S A268719 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 A268719 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 A268719 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 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), ...
%H A268719 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 A268719 T(n,k) = A003188(A006068(n) + A006068(k)).
%F A268719 a(n) = A268715(A003056(n), A002262(n)). [As a linear sequence.]
%e A268719 The first fifteen rows of the triangle:
%e A268719                              0
%e A268719                            1   3
%e A268719                          2   6   5
%e A268719                        3   2   7   6
%e A268719                      4  12  15  13   9
%e A268719                    5   4  13  12  11  10
%e A268719                  6   7   4   5  14  15  12
%e A268719                7   5  12   4  10  14  13  15
%e A268719              8  24  27  25  29  31  26  30  17
%e A268719            9   8  25  24  31  30  27  26  19  18
%e A268719         10  11   8   9  26  27  24  25  22  23  20
%e A268719       11   9  24   8  30  26  25  27  18  22  21  23
%e A268719     12  13  14  15   8   9  10  11  28  29  30  31  24
%e A268719   13  15  10  14  24   8  11   9  20  28  31  29  25  27
%e A268719 14  10   9  11  27  25   8  24  23  21  28  20  26  30  29
%t A268719 a88[n_] := BitXor[n, Floor[n/2]];
%t A268719 a68[n_] := BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}];
%t A268719 a68[0] = 0;
%t A268719 T[n_, k_] := a88[a68[n] + a68[k]];
%t A268719 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 19 2019 *)
%o A268719 (Scheme) (define (A268719 n) (A268715bi (A003056 n) (A002262 n)))
%o A268719 (Python)
%o A268719 def a003188(n): return n^(n>>1)
%o A268719 def a006068(n):
%o A268719     s=1
%o A268719     while True:
%o A268719         ns=n>>s
%o A268719         if ns==0: break
%o A268719         n=n^ns
%o A268719         s<<=1
%o A268719     return n
%o A268719 def T(n, k): return a003188(a006068(n) + a006068(k))
%o A268719 for n in range(21): print([T(n, k) for k in range(n + 1)]) # _Indranil Ghosh_, Jun 07 2017
%Y A268719 Cf. A003188, A006068.
%Y A268719 Cf. A002262, A003056, A268715.
%Y A268719 Cf. A001477 (left edge), A001969 (right edge).
%Y A268719 Cf. A268720 (row sums).
%K A268719 nonn,tabl
%O A268719 0,3
%A A268719 _Antti Karttunen_, Feb 13 2016