A268720 -id:A268720 - 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), ...

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

Antti Karttunen, Feb 13 2016

			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

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

Cf. A003188, A006068.
Cf. A002262, A003056, A268715.
Cf. A001477 (left edge), A001969 (right edge).
Cf. A268720 (row sums).

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 *)

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) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268719 n) (A268715bi (A003056 n) (A002262 n)))

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

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

A268837 Antidiagonal sums of array A268715: a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

Original entry on oeis.org

0, 2, 7, 18, 17, 48, 56, 80, 67, 122, 136, 194, 204, 268, 281, 328, 291, 378, 396, 498, 510, 640, 675, 792, 790, 886, 965, 1098, 1093, 1208, 1248, 1344, 1227, 1378, 1356, 1530, 1538, 1792, 1815, 2016, 2008, 2218, 2339, 2602, 2619, 2892, 2970, 3208, 3150, 3294, 3385, 3586, 3691, 4012, 4174, 4440, 4367, 4554, 4644

Offset: 0

Antti Karttunen, Feb 15 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A003188, A006068.

Cf. also A268720, A268836.

(define (A268837 n) (add (lambda (k) (A003188 (+ (A006068 k) (A006068 (- n k))))) 0 n))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k=0..n} A003188(A006068(n)+A006068(n-k)).

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), ...

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