A224924 Sum_{i=0..n} Sum_{j=0..n} (i AND j), where AND is the binary logical AND operator.

0, 1, 3, 12, 16, 33, 63, 112, 120, 153, 211, 300, 408, 553, 735, 960, 976, 1041, 1155, 1324, 1536, 1809, 2143, 2544, 2952, 3433, 3987, 4620, 5320, 6105, 6975, 7936, 7968, 8097, 8323, 8652, 9072, 9601, 10239, 10992, 11800, 12729, 13779, 14956, 16248, 17673, 19231, 20928

Offset

0,3

Comments

For n>0, a(2^n)-A000217(2^n)=a(2^n-1)-A000217(2^n-1) [See links]. - R. J. Cano, Aug 21 2013

Links

Enrique Pérez Herrero, Table of n, a(n) for n = 0..1000

R. J. Cano, Additional information

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 42-43.

Formula

a(2^n) = a(2^n - 1) + 2^n.

a(n) -a(n-1) = 2*A222423(n) -n. - R. J. Mathar, Aug 22 2013

Maple

read("transforms") :
A224924 := proc(n)
    local a, i, j ;
    a := 0 ;
    for i from 0 to n do
    for j from 0 to n do
        a := a+ANDnos(i, j) ;
    end do:
    end do:
    a ;
end proc: # R. J. Mathar, Aug 22 2013

Mathematica

a[n_] := Sum[BitAnd[i, j], {i, 0, n}, {j, 0, n}];
Table[a[n], {n, 0, 20}]
(* Enrique Pérez Herrero, May 30 2015 *)

Prog

(Python)
for n in range(99):
    s = 0
    for i in range(n+1):
      for j in range(n+1):
        s += i & j
    print(s, end=', ')
(PARI) a(n)=sum(i=0, n, sum(j=0, n, bitand(i, j))); \\ R. J. Cano, Aug 21 2013

Crossrefs

Cf. A004125, A222423, A224915, A224923, A000217.

Sequence in context: A064106 A022411 A115229 * A136047 A082965 A045549

Adjacent sequences: A224921 A224922 A224923 * A224925 A224926 A224927

Keyword

nonn,base

Author

Alex Ratushnyak, Apr 19 2013

Status

approved