A080099 -id:A080099 - cp's OEIS frontend

A080100 a(n) = 2^(number of 0's in binary representation of n).

1, 1, 2, 1, 4, 2, 2, 1, 8, 4, 4, 2, 4, 2, 2, 1, 16, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1, 32, 16, 16, 8, 16, 8, 8, 4, 16, 8, 8, 4, 8, 4, 4, 2, 16, 8, 8, 4, 8, 4, 4, 2, 8, 4, 4, 2, 4, 2, 2, 1, 64, 32, 32, 16, 32, 16, 16, 8, 32, 16, 16, 8, 16, 8, 8, 4, 32, 16, 16, 8, 16, 8, 8, 4, 16, 8, 8, 4, 8, 4

Offset: 0

Reinhard Zumkeller, Jan 28 2003

Number of numbers k, 0<=k<=n, such that (k AND n) = 0 (bitwise logical AND): a(n) = #{k : T(n,k)=n, 0<=k<=n}, where T is defined as in A080099.

Same parity as the Catalan numbers (A000108). - Paul D. Hanna, Nov 14 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Cf. A001316.
Cf. A002487.
This is Guy Steele's sequence GS(5, 3) (see A135416).
Cf. A048896.
Cf. A023416, A080791.

import Data.List (transpose)
a080100 n = a080100_list !! n
a080100_list =  1 : zs where
   zs =  1 : (concat $ transpose [map (* 2) zs, zs])
-- Reinhard Zumkeller, Aug 27 2014, Mar 07 2011

a:= n-> 2^add(1-i, i=Bits[Split](n)):
seq(a(n), n=0..93);  # Alois P. Heinz, Aug 18 2025

f[n_] := 2^DigitCount[n, 2, 0]; f[0] = 1; Array[f, 94, 0] (* Robert G. Wilson v *)

PARI
```
a(n)=if(n<1,n==0,(2-n%2)*a(n\2))
```

a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,x^2)*(2+x)-1); polcoeff(A,n))

def A080100(n): return 1<Chai Wah Wu, Aug 18 2025

G.f. satisfies: F(x^2) = (1+F(x))/(x+2). - Ralf Stephan, Jun 28 2003
a(2n) = 2a(n), n>0. a(2n+1) = a(n). - Ralf Stephan, Apr 29 2003
a(n) = 2^A080791(n). a(n)=2^A023416(n), n>0.
a(n) = sum(k=0, n, C(n+k, k) mod 2). - Benoit Cloitre, Mar 06 2004
a(n) = sum(k=0, n, C(2n-k, n) mod 2). - Paul Barry, Dec 13 2004
G.f. satisfies: A(x) = Sum_{n>=0} [A(x)^n (mod 2)]*x^n, where A(x)^n (mod 2) reduces all coefficients modulo 2 to {0,1}. - Paul D. Hanna, Nov 14 2012

Keyword base added by Rémy Sigrist, Jan 18 2018

A265705 Triangle read by rows: T(n,k) = k IMPL n, 0 <= k <= n, bitwise logical IMPL.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Dec 15 2015

			.          10 | 1010                            12 | 1100
.           4 |  100                             6 |  110
.   ----------+-----                     ----------+-----
.   4 IMPL 10 | 1011 -> T(10,4)=11       6 IMPL 12 | 1101 -> T(12,6)=13
.
First 16 rows of the triangle, where non-symmetrical rows are marked, see comment concerning A158582 and A089633:
.   0:                                 0
.   1:                               1   1
.   2:                             3   2   3
.   3:                           3   3   3   3
.   4:                         7   6   5   4   7    X
.   5:                       7   7   5   5   7   7
.   6:                     7   6   7   6   7   6   7
.   7:                   7   7   7   7   7   7   7   7
.   8:                15  14  13  12  11  10   9   8  15    X
.   9:              15  15  13  13  11  11   9   9  15  15    X
.  10:            15  14  15  14  11  10  11  10  15  14  15    X
.  11:          15  15  15  15  11  11  11  11  15  15  15  15
.  12:        15  14  13  12  15  14  13  12  15  14  13  12  15    X
.  13:      15  15  13  13  15  15  13  13  15  15  13  13  15  15
.  14:    15  14  15  14  15  14  15  14  15  14  15  14  15  14  15
.  15:  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15 .

Reinhard Zumkeller, Rows n = 0..255 of triangle, flattened
Eric Weisstein's World of Mathematics, Implies

Cf. A003817, A007088, A029578, A089633, A158582, A247648, A265716 (central terms), A265736 (row sums).
Cf. A053644, A265885, A327490.
Other triangles: A080099 (AND), A080098 (OR), A051933 (XOR), A102037 (CNIMPL).

a265705_tabl = map a265705_row [0..]
a265705_row n = map (a265705 n) [0..n]
a265705 n k = k `bimpl` n where
   bimpl 0 0 = 0
   bimpl p q = 2 * bimpl p' q' + if u <= v then 1 else 0
               where (p', u) = divMod p 2; (q', v) = divMod q 2

using IntegerSequences
for n in 0:15 println(n == 0 ? [0] : [Bits("IMP", k, n) for k in 0:n]) end  # Peter Luschny, Sep 25 2021

A265705 := (n, k) -> Bits:-Implies(k, n):
seq(seq(A265705(n, k), k=0..n), n=0..11); # Peter Luschny, Sep 23 2019

T[n_, k_] := If[n == 0, 0, BitOr[2^Length[IntegerDigits[n, 2]]-1-k, n]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 25 2021, after David A. Corneth's PARI code *)

T(n, k) = if(n==0,return(0)); bitor((2<David A. Corneth, Sep 24 2021

T(n,0) = T(n,n) = A003817(n).
T(2*n,n) = A265716(n).
Let m = A089633(n): T(m,k) = T(m,m-k), k = 0..m.
Let m = A158582(n): T(m,k) != T(m,m-k) for at least one k <= n.
Let m = A247648(n): T(2*m,m) = 2*m.
For n > 0: A029578(n+2) = number of odd terms in row n; no even terms in odd-indexed rows.
A265885(n) = T(prime(n),n).
A053644(n) = smallest k such that row k contains n.

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

Original entry on oeis.org

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

A213370 a(n) = n AND 2*n, where AND is the bitwise AND operator.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 4, 6, 0, 0, 0, 2, 8, 8, 12, 14, 0, 0, 0, 2, 0, 0, 4, 6, 16, 16, 16, 18, 24, 24, 28, 30, 0, 0, 0, 2, 0, 0, 4, 6, 0, 0, 0, 2, 8, 8, 12, 14, 32, 32, 32, 34, 32, 32, 36, 38, 48, 48, 48, 50, 56, 56, 60, 62, 0, 0, 0, 2, 0, 0, 4, 6, 0, 0, 0, 2, 8, 8

Offset: 0

Alex Ratushnyak, Jun 14 2012

Antti Karttunen, Table of n, a(n) for n = 0..16384 (terms 0..1023 from T. D. Noe)
Index entries for sequences related to binary expansion of n

Cf. A080099, A213526, A048727, A048735, A269160.
Cf. A003714: indices of 0's.
Cf. A213540: indices of 2's, indices of 4's divided by 2.

Table[BitAnd[n, 2n], {n, 0, 63}] (* Alonso del Arte, Jun 19 2012 *)

a(n) = bitand(n, 2*n); \\ Michel Marcus, Mar 26 2021

for n in range(99):
    print(2*n & n, end=", ")

a(n) = 2 * A048735(n).

a(n) = (1/2)*(A048727(n) XOR A269160(n)) = (n OR 2n) XOR (n XOR 2n). - Antti Karttunen, May 16 2021

A051933 Triangle T(n,m) = Nim-sum (or XOR) of n and m, read by rows, 0<=m<=n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 20 1999

			{0},
{1,0},
{2,3,0},
{3,2,1,0}, ...

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 60.
J. H. Conway, On Numbers and Games, Academic Press, p. 52.

Reinhard Zumkeller, Rows n = 0..127 of triangle, flattened first 50 rows by R. J. Mathar
Index entries for sequences related to Nim-sums

Cf. A224915 (row sums), A003987 (array), A051910 (Nim-product).

Other triangles: A080099 (AND), A080098 (OR), A265705 (IMPL), A102037 (CNIMPL), A002262 (k).

import Data.Bits (xor)
a051933 n k = n `xor` k :: Int
a051933_row n = map (a051933 n) [0..n]
a051933_tabl = map a051933_row [0..]
-- Reinhard Zumkeller, Aug 02 2014, Aug 13 2013

using IntegerSequences
A051933Row(n) = [Bits("XOR", n, k) for k in 0:n]
for n in 0:10 println(A051933Row(n)) end  # Peter Luschny, Sep 25 2021

nimsum := proc(a,b) local t1,t2,t3,t4,l; t1 := convert(a+2^20,base,2); t2 := convert(b+2^20,base,2); t3 := evalm(t1+t2); map(x->x mod 2, t3); t4 := convert(evalm(%),list); l := convert(t4,base,2,10); sum(l[k]*10^(k-1), k=1..nops(l)); end; # memo: adjust 2^20 to be much bigger than a and b
AT := array(0..N,0..N); for a from 0 to N do for b from a to N do AT[a,b] := nimsum(a,b); AT[b,a] := AT[a,b]; od: od:
# Alternative:
A051933 := (n, k) -> Bits:-Xor(n, k):
seq(seq(A051933(n, k), k=0..n), n=0..12); # Peter Luschny, Sep 23 2019

Flatten[Table[BitXor[m, n], {m, 0, 12}, {n, 0, m}]] (* Jean-François Alcover, Apr 29 2011 *)

More terms from Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

A222423 Sum of (n AND k) for k = 0, 1, 2, ..., n, where AND is the bitwise AND operator.

Original entry on oeis.org

0, 1, 2, 6, 4, 11, 18, 28, 8, 21, 34, 50, 60, 79, 98, 120, 16, 41, 66, 94, 116, 147, 178, 212, 216, 253, 290, 330, 364, 407, 450, 496, 32, 81, 130, 182, 228, 283, 338, 396, 424, 485, 546, 610, 668, 735, 802, 872, 816, 889, 962, 1038, 1108, 1187, 1266, 1348, 1400

Offset: 0

Alex Ratushnyak, Feb 23 2013

If n = 2^x, (n AND k) = 0 for k < n, therefore a(n) = n if and only if n = 0 or n = 2^x.
Row sums of A080099. - R. J. Mathar, Apr 26 2013
The associated incomplete sum_{0<=kA213673(n). - R. J. Mathar, Aug 22 2013

			a(3) = 6 because 1 AND 3 = 1; 2 AND 3 = 2; 3 AND 3 = 3; and 1 + 2 + 3 = 6.
a(4) = 4 because 1 AND 4 = 0; 2 AND 4 = 0; 3 AND 4 = 0; 4 AND 4 = 4; and 0 + 0 + 0 + 4 = 4.
a(5) = 11 because 1 AND 5 = 1; 2 AND 5 = 0; 3 AND 5 = 1; 4 AND 5 = 4; 5 AND 5 = 5; and 1 + 0 + 1 + 4 + 5 = 11.

Ivan Neretin, Table of n, a(n) for n = 0..8192

Cf. A004125.

Table[Sum[BitAnd[n, k], {k, 0, n}], {n, 0, 63}] (* Alonso del Arte, Feb 24 2013 *)

a(n) = sum(k=0, n, bitand(n, k)); \\ Michel Marcus, May 17 2015

for n in range(99):
    s = 0
    for k in range(n+1):
        s += n & k
    print(s, end=",")

a(2^n-1) = A006516(n) for all n, since k AND 2^n-1 = k for all k<2^n. - M. F. Hasler, Feb 28 2013

A102210 Number of primes that are bitwise covered by n.

Original entry on oeis.org

0, 1, 2, 0, 1, 1, 4, 0, 0, 1, 3, 0, 2, 1, 6, 0, 1, 1, 4, 0, 2, 1, 7, 0, 1, 1, 5, 0, 4, 1, 11, 0, 0, 1, 2, 0, 2, 1, 5, 0, 1, 1, 5, 0, 4, 1, 10, 0, 1, 1, 4, 0, 4, 1, 9, 0, 2, 1, 8, 0, 8, 1, 18, 0, 0, 1, 3, 0, 1, 1, 6, 0, 1, 1, 5, 0, 3, 1, 10, 0, 1, 1, 6, 0, 2, 1, 10, 0, 3, 1, 9, 0, 6, 1, 17, 0, 1, 1, 4, 0, 4, 1

Offset: 1

Reinhard Zumkeller, Dec 30 2004

p is bitwise covered by n iff (p = (n AND p)) bitwise: A080099(n,p)=p.

			n=21->10101 -> a(21) = #{00101=5,10001=17} = 2.

Cf. A007053, A007088, A004676, A080099, A102211, A102212, A102213, A102550.

[#[p:p in PrimesUpTo(n)| p eq BitwiseAnd(n,p)] :n in [1..105] ]; // Marius A. Burtea, Jan 12 2020

a[n_] := Count[Range[n], ?(PrimeQ[#] && BitAnd[n, #] == # &)]; Array[a, 100] (* _Amiram Eldar, Jan 12 2020 *)

a(A102211(n)) = 0; a(A102212(n)) = 1; a(A102213(n)) > 1.

a(2^k-1) = A007053(k) for k > 1. - Amiram Eldar, Jan 12 2020

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

A102550 Number of distinct prime-factors of n that are bitwise covered by n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0

Offset: 1

Reinhard Zumkeller, Jan 14 2005

nonn

p is bitwise covered by n iff (p = (n AND p)) bitwise: A080099(n,p)=p.

Index entries for sequences related to binary expansion of n

Cf. A001221 (omega), A007088, A004676, A080099, A102210.

Cf. A102551, A102553, A102554, A102555.

a[1] = 0; a[k_] := Module[{f=FactorInteger[k][[;; , 1]]}, Count[BitAnd[k, f]-f, 0]];  Array[a,120] (* Amiram Eldar, Feb 06 2019 *)

a(A102553(n)) = A001221(A102553(n));
a(A102554(n)) < A001221(A102554(n));
a(A102551(n)) = 0, a(A102551(n)) > 0;
a(A102555(n)) = n;
a(m) < n for m < A102555(n).
a(n) = Sum_{p|n} (binomial(n,p) mod 2), where p is a prime. - Ridouane Oudra, May 03 2019

Offset 1 from Amiram Eldar, Feb 06 2019

A327853 Triangle read by rows, Sierpinski's gasket, A047999 * (0,1,2,3,4,...) diagonalized.

Original entry on oeis.org

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

Offset: 1

Matej Veselovac, Sep 28 2019

This is similar to A166555, the difference being that this is scaled "linearly" instead of exponentially.
The scatterplot of the sequence resembles Sierpinski's gasket (triangle), with a square root border (the "linear" scaling is not normalized and actually resembles the scale of the function of the positive inverse of triangular numbers: A003056).
If instead of (0,1,2,3,4,...), we use the A000217 (triangular numbers), then the border of the scatterplot will be truly linear.

			First 16 rows of the triangle:
  0;
  0, 1;
  0, 0, 2;
  0, 1, 2, 3;
  0, 0, 0, 0, 4;
  0, 1, 0, 0, 4, 5;
  0, 0, 2, 0, 4, 0, 6;
  0, 1, 2, 3, 4, 5, 6, 7;
  0, 0, 0, 0, 0, 0, 0, 0, 8;
  0, 1, 0, 0, 0, 0, 0, 0, 8, 9;
  0, 0, 2, 0, 0, 0, 0, 0, 8, 0, 10;
  0, 1, 2, 3, 0, 0, 0, 0, 8, 9, 10, 11;
  0, 0, 0, 0, 4, 0, 0, 0, 8, 0,  0,  0, 12;
  0, 1, 0, 0, 4, 5, 0, 0, 8, 9,  0,  0, 12, 13;
  0, 0, 2, 0, 4, 0, 6, 0, 8, 0, 10,  0, 12,  0, 14;
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15;

Matej Veselovac, Table of n, a(n) for n = 1..100000
Math StackExchange, Pattern in Pascal's triangle .
Matej Veselovac, Scatterplot and Logarithmic Scatterplot of triangle read by rows T(n,k) = k * (binomial(n, k) mod 2), n=1...10^5..

Cf. A001317, A007318, A003056, A000217.
Cf. A166555 (2^k is used instead of k).
Cf. A080099 (similar scatterplot visualization).
Cf. A327889 (alternating, normalized (linear) modification of the sequence, transformed by first decimal digit indicator function).

r[n0_]:=Flatten[Table[(k)(Mod[Binomial[n,k],2]),{n,0,n0},{k,0,n}]]; r[20] (* Matej Veselovac, Sep 28 2019 *)

Triangle read by rows, A047999 * Q. A047999 = Sierpinski's gasket, Q = an infinite lower triangular matrix with (0,1,2,3,...) as the main diagonal and the rest zeros.

The entries of the triangle are given by T(n, k) = k * (binomial(n, k) (mod 2)), then it is read by rows.

cp's OEIS Frontend

A080100 a(n) = 2^(number of 0's in binary representation of n).

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A265705 Triangle read by rows: T(n,k) = k IMPL n, 0 <= k <= n, bitwise logical IMPL.

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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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A213370 a(n) = n AND 2*n, where AND is the bitwise AND operator.

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A051933 Triangle T(n,m) = Nim-sum (or XOR) of n and m, read by rows, 0<=m<=n.

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A222423 Sum of (n AND k) for k = 0, 1, 2, ..., n, where AND is the bitwise AND operator.

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