A350094 -id:A350094 - cp's OEIS frontend

A224915 a(n) = Sum_{k=0..n} n XOR k where XOR is the bitwise logical exclusive-or operator.

0, 1, 5, 6, 22, 23, 27, 28, 92, 93, 97, 98, 114, 115, 119, 120, 376, 377, 381, 382, 398, 399, 403, 404, 468, 469, 473, 474, 490, 491, 495, 496, 1520, 1521, 1525, 1526, 1542, 1543, 1547, 1548, 1612, 1613, 1617, 1618, 1634, 1635, 1639, 1640, 1896, 1897, 1901, 1902, 1918

Offset: 0

Alex Ratushnyak, Apr 19 2013

			a(2) = (0 xor 2) + (1 xor 2) = 2 + 3 = 5.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A001196 (bit doubling).
Row sums of A051933.
Other sums: A222423 (AND), A350093 (OR), A265736 (IMPL), A350094 (CNIMPL), A004125 (mod).

read("transforms"):
A051933 := proc(n,k)
    XORnos(n,k) ;
end proc:
A224915 := proc(n)
    add(A051933(n,k),k=0..n) ;
end proc: # R. J. Mathar, Apr 26 2013
# second Maple program:
with(MmaTranslator[Mma]):
seq(add(BitXor(n,i),i=0..n),n=0..60); # Ridouane Oudra, Dec 09 2020

Array[Sum[BitXor[#, k], {k, 0, #}] &, 53, 0] (* Michael De Vlieger, Dec 09 2020 *)

a(n) = sum(k=0, n, bitxor(n, k)); \\ Michel Marcus, Jun 08 2019

a(n) = (3*fromdigits(binary(n),4) - n) >>1; \\ Kevin Ryde, Dec 17 2021

for n in range(59):
    s = 0
    for k in range(n):  s += n ^ k
    print(s, end=',')

def A224915(n): return 3*int(bin(n)[2:],4)-n>>1 # Chai Wah Wu, Aug 21 2023

a(n) = Sum_{j=1..n} 4^(v_2(j)), where v_2(j) is the exponent of highest power of 2 dividing j. - Ridouane Oudra, Jun 08 2019
a(n) = n + 3*Sum_{j=1..floor(log_2(n))} 4^(j-1)*floor(n/2^j), for n>=1. - Ridouane Oudra, Dec 09 2020
From Kevin Ryde, Dec 17 2021: (Start)
a(2*n+b) = 4*a(n) + n + b where b = 0 or 1.
a(n) = (A001196(n) - n)/2.
a(n) = A350093(n) - A222423(n), being XOR = OR - AND.
(End)

A102037 Triangle of BitAnd(BitNot(n), k).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 3, 0, 0, 0, 2, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 2, 2, 4, 4, 6, 6, 0, 0, 0, 1, 0, 1, 4, 5, 4, 5, 0, 1, 0, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0

Offset: 0

Eric W. Weisstein, Dec 25 2004

As a logical operation on two variables this is also called the 'converse nonimplication'. - Peter Luschny, Sep 25 2021

			Table starts:
[0] 0;
[1] 0, 0;
[2] 0, 1, 0;
[3] 0, 0, 0, 0;
[4] 0, 1, 2, 3, 0;
[5] 0, 0, 2, 2, 0, 0;
[6] 0, 1, 0, 1, 0, 1, 0;
[7] 0, 0, 0, 0, 0, 0, 0, 0;
[8] 0, 1, 2, 3, 4, 5, 6, 7, 0;
[9] 0, 0, 2, 2, 4, 4, 6, 6, 0, 0.

Eric Weisstein's World of Mathematics, Sierpinski Sieve
Wikipedia, Converse nonimplication.

Cf. A350094 (row sums), A268040 (array).

Other triangles: A080099 (AND), A080098 (OR), A051933 (XOR), A265705 (IMPL).

using IntegerSequences
A102037Row(n) = [Bits("CNIMP", n, k) for k in 0:n]
for n in 0:20 println(A102037Row(n)) end  # Peter Luschny, Sep 25 2021

with(Bits): cnimp := (n, k) -> And(Not(n), k):
seq(print(seq(cnimp(n,k), k=0..n)), n = 0..12); # Peter Luschny, Sep 25 2021

A350093 a(n) = Sum_{k=0..n} n OR k where OR is the bitwise logical OR operator (A003986).

Original entry on oeis.org

0, 2, 7, 12, 26, 34, 45, 56, 100, 114, 131, 148, 174, 194, 217, 240, 392, 418, 447, 476, 514, 546, 581, 616, 684, 722, 763, 804, 854, 898, 945, 992, 1552, 1602, 1655, 1708, 1770, 1826, 1885, 1944, 2036, 2098, 2163, 2228, 2302, 2370, 2441, 2512, 2712, 2786, 2863

Offset: 0

Kevin Ryde, Dec 14 2021

The effect of n OR k is to force a 1-bit at all bit positions where n has a 1-bit, which means n*(n+1) in the sum. Bits of k where n has a 0-bit are NOT(n) AND k = n CNIMPL k so that a(n) = A350094(n) + n*(n+1).

Winston de Greef, Table of n, a(n) for n = 0..10000

Cf. A003986 (bitwise OR), A001196 (bit doubling).
Row sums of A080098.
Other sums: A222423 (AND), A224915 (XOR), A265736 (IMPL), A350094 (CNIMPL).

a(n) = (3*(n^2 + fromdigits(binary(n),4)) + 2*n) >> 2;

a(n) = ((3*n+2)*n + A001196(n)) / 4.
a(2*n) = 4*a(n) - n.
a(2*n+1) = 4*a(n) + 2*n + 2.
a(n) = A222423(n) + A224915(n), being OR = AND + XOR.

cp's OEIS Frontend

A224915 a(n) = Sum_{k=0..n} n XOR k where XOR is the bitwise logical exclusive-or operator.

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A102037 Triangle of BitAnd(BitNot(n), k).

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A350093 a(n) = Sum_{k=0..n} n OR k where OR is the bitwise logical OR operator (A003986).

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PARI

Formula