A350093 -id:A350093 - cp's OEIS frontend

A080098 Triangle T(n,k) = n OR k, 0 <= k <= n, bitwise logical OR, read by rows.

0, 1, 1, 2, 3, 2, 3, 3, 3, 3, 4, 5, 6, 7, 4, 5, 5, 7, 7, 5, 5, 6, 7, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 9, 11, 11, 13, 13, 15, 15, 9, 9, 10, 11, 10, 11, 14, 15, 14, 15, 10, 11, 10, 11, 11, 11, 11, 15, 15, 15, 15, 11, 11, 11, 11, 12, 13, 14, 15, 12, 13, 14, 15, 12, 13, 14, 15, 12

Offset: 0

Reinhard Zumkeller, Jan 28 2003

			Triangle begins:
   0,
   1,  1,
   2,  3,  2,
   3,  3,  3,  3,
   4,  5,  6,  7,  4,
   5,  5,  7,  7,  5,  5,
   6,  7,  6,  7,  6,  7,  6,
   7,  7,  7,  7,  7,  7,  7,  7,
   8,  9, 10, 11, 12, 13, 14, 15,  8,
   9,  9, 11, 11, 13, 13, 15, 15,  9,  9,
  10, 11, 10, 11, 14, 15, 14, 15, 10, 11, 10,
  ...

Rick L. Shepherd, Rows n = 0..500 of triangle, flattened
Eric Weisstein's World of Mathematics, OR.

Cf. A001316 (number of integers k such that T(n, k) = n in n-th row).
Cf. A350093 (row sums), A003986 (array).
Other triangles: A080099 (AND), A051933 (XOR), A265705 (IMPL), A102037 (CNIMPL).

import Data.Bits ((.|.))
a080098 n k = n .|. k :: Int
a080098_row n = map (a080098 n) [0..n]
a080098_tabl = map a080098_row [0..]
-- Reinhard Zumkeller, Aug 03 2014, Jul 05 2012

T[n_, k_] := n ~BitOr~ k;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)

def T(n, k): return n | k
print([T(n, k) for n in range(13) for k in range(n+1)]) # Michael S. Branicky, Dec 01 2021

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

Original entry on oeis.org

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)

A350094 a(n) = Sum_{k=0..n} n CNIMPL k where CNIMPL = NOT(n) AND k is the bitwise logical converse non-implication operator (A102037).

Original entry on oeis.org

0, 0, 1, 0, 6, 4, 3, 0, 28, 24, 21, 16, 18, 12, 7, 0, 120, 112, 105, 96, 94, 84, 75, 64, 84, 72, 61, 48, 42, 28, 15, 0, 496, 480, 465, 448, 438, 420, 403, 384, 396, 376, 357, 336, 322, 300, 279, 256, 360, 336, 313, 288, 270, 244, 219, 192, 196, 168, 141, 112

Offset: 0

Kevin Ryde, Dec 14 2021

The effect of NOT(n) AND k is to retain from k only those bits where n has a 0-bit. Conversely n AND k retains from k those bits where n has a 1-bit. Together they are all bits of k so that a(n) + A222423(n) = Sum_{k=0..n} k = n*(n+1)/2.

Row sums of A102037.
Cf. A001196 (bit doubling).
Other sums: A222423 (AND), A350093 (OR), A224915 (XOR), A265736 (IMPL).

with(Bits): cnimp := (n, k) -> And(Not(n), k):
seq(add(cnimp(n, k), k = 0..n), n = 0..59); # Peter Luschny, Dec 14 2021

a(n) = (3*fromdigits(binary(n),4) - n^2 - 2*n)/4;

a(n) = (A001196(n) - n*(n+2))/4.
a(2*n) = 4*a(n) + n.
a(2*n+1) = 4*a(n).

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A080098 Triangle T(n,k) = n OR k, 0 <= k <= n, bitwise logical OR, read by rows.

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

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A350094 a(n) = Sum_{k=0..n} n CNIMPL k where CNIMPL = NOT(n) AND k is the bitwise logical converse non-implication operator (A102037).

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