A213370 -id:A213370 - cp's OEIS frontend

A163617 a(2n) = 2a(n), a(2n + 1) = 2a(n) + 2 + (-1)^n, for all n in Z.

0, 3, 6, 7, 12, 15, 14, 15, 24, 27, 30, 31, 28, 31, 30, 31, 48, 51, 54, 55, 60, 63, 62, 63, 56, 59, 62, 63, 60, 63, 62, 63, 96, 99, 102, 103, 108, 111, 110, 111, 120, 123, 126, 127, 124, 127, 126, 127, 112, 115, 118, 119, 124, 127, 126, 127, 120, 123, 126, 127, 124, 127, 126

Offset: 0

Michael Somos, Aug 01 2009

nonn

Fibbinary numbers (A003714) give all integers n >= 0 for which a(n) = 3*n.
From Antti Karttunen, Feb 21 2016: (Start)
Fibbinary numbers also give all integers n >= 0 for which a(n) = A048724(n).
Note that there are also other multiples of three in the sequence, for example, A163617(99) = 231 ("11100111" in binary) = 3*77, while 77 ("1001101" in binary) is not included in A003714. Note that 99 is "1100011" in binary.
(End)

			G.f. = 3*x + 6*x^2 + 7*x^3 + 12*x^4 + 15*x^5 + 14*x^6 + 15*x^7 + 24*x^8 + 27*x^9 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A003986, A048724, A213370, A163618.

Cf. also A269161.

import Data.Bits ((.|.), shiftL)
a163617 n = n .|. shiftL n 1 :: Integer
-- Reinhard Zumkeller, Mar 06 2013

using IntegerSequences
A163617List(len) = [Bits("OR", n, n<<1) for n in 0:len]
println(A163617List(62))  # Peter Luschny, Sep 26 2021

A163617 := n -> Bits:-Or(2*n, n):
seq(A163617(n), n=0..62); # Peter Luschny, Sep 23 2019

Table[BitOr[n, 2*n], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)

PARI
```
{a(n) = bitor(n, n<<1)};
```

{a(n) = if( n==0 || n==-1, n, 2 * a(n \ 2) + (n%2) * (2 + (-1)^(n \ 2)))};

(define (A163617 n) (A003986bi n (+ n n))) ;; Here A003986bi implements dyadic bitwise-OR operation (see A003986) - Antti Karttunen, Feb 21 2016

a(n) = -A163618(-n) for all n in ZZ.

Conjecture: a(n) = A003188(n) + (6*n + 1 - (-1)^n)/4. - Velin Yanev, Dec 17 2016

Comment about Fibbinary numbers rephrased by Antti Karttunen, Feb 21 2016

A213541 a(n) = n AND n^2, where AND is the bitwise AND operator.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 4, 1, 0, 1, 0, 9, 0, 9, 4, 1, 0, 1, 0, 1, 16, 17, 4, 17, 0, 17, 0, 25, 16, 9, 4, 1, 0, 1, 0, 1, 0, 1, 36, 33, 0, 1, 32, 41, 0, 41, 4, 33, 0, 33, 0, 33, 16, 49, 36, 17, 0, 49, 32, 25, 16, 9, 4, 1, 0, 1, 0, 1, 0, 1, 4, 1, 64, 65, 64, 73, 0, 9, 68

Offset: 0

Alex Ratushnyak, Jun 14 2012

The graph of this sequence has the shape of a tilted Sierpinski triangle. - WG Zeist, Jan 15 2019

Reinhard Zumkeller, Table of n, a(n) for n = 0..8192

Cf. A213370.
Cf. A000290.
Cf. A007745 (OR), A169810 (XOR), A002378.

import Data.Bits ((.&.))
a213541 n = n .&. n ^ 2  -- Reinhard Zumkeller, Apr 25 2013

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

a(n) = bitand(n, n^2); \\ Michel Marcus, Jan 15 2019

Python
```
print([n*n & n for n in range(99)])
```

a(2^k + x) = a(x) + (x^2 AND 2^k) for 0 <= x < 2^k. - David Radcliffe, May 06 2023

A292373 A binary encoding of 3-digits in base-4 representation of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0          1           0
   2        0          2           0
   3        1          3           1
   4        0         10           0
   5        0         11           0
   6        0         12           0
   7        1         13           1
   8        0         20           0
   9        0         21           0
  10        0         22           0
  11        1         23           1
  12        2         30          10
  13        2         31          10
  14        2         32          10
  15        3         33          11
  16        0        100           0
  17        0        101           0
  18        0        102           0
  19        1        103           1

Antti Karttunen, Table of n, a(n) for n = 0..65536
Index entries for sequences related to binary expansion of n

Cf. A007088, A007090, A048735, A059905, A059906, A160383, A213370, A292370, A292371, A292372.

def A292373(n): return int(bin(n&n>>1)[:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059905(A048735(n)) = A059906(A213370(n)).

For all n >= 0, A000120(a(n)) = A160383(n).

A213540 Numbers k such that k AND k*2 = 2, where AND is the bitwise AND operator.

Original entry on oeis.org

3, 11, 19, 35, 43, 67, 75, 83, 131, 139, 147, 163, 171, 259, 267, 275, 291, 299, 323, 331, 339, 515, 523, 531, 547, 555, 579, 587, 595, 643, 651, 659, 675, 683, 1027, 1035, 1043, 1059, 1067, 1091, 1099, 1107, 1155, 1163, 1171, 1187, 1195, 1283, 1291, 1299, 1315

Offset: 1

Alex Ratushnyak, Jun 14 2012

			In binary, 19 is 10011, while 2 * 19 = 38 is of course 100110. Since 010011 AND 100110 = 000010 (in decimal, 2), 19 is in the sequence.
20 is not in the sequence, since 010100 AND 101000 = 000000.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A213370, A004198, A003714.

F:= combinat[fibonacci]:
b:= proc(n) local j;
      if n=0 then 0
    else for j from 2 while F(j+1)<=n do od;
         b(n-F(j))+2^(j-2)
      fi
    end:
a:= n-> 8*b(n-1)+3:
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 17 2012

Select[Range[1024], BitAnd[#, 2#] == 2 &] (* Alonso del Arte, Jun 18 2012 *)

is(n)=bitand(n,2*n)==2 \\ Charles R Greathouse IV, Jun 18 2012

for n in range(1777):
    a = 2*n & n
    if a==2:
        print(n, end=',')

a(n) = A003714(n-1)*8 + 3.

A215488 a(0)=1, a(n) = a(n-1) + a(2*n AND n), where AND is the bitwise AND operator.

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 15, 30, 31, 32, 33, 36, 67, 98, 165, 330, 331, 332, 333, 336, 337, 338, 345, 360, 691, 1022, 1353, 1686, 2377, 3068, 5445, 10890, 10891, 10892, 10893, 10896, 10897, 10898, 10905, 10920, 10921, 10922, 10923, 10926, 10957, 10988, 11055, 11220, 22111, 33002

Offset: 0

Alex Ratushnyak, Aug 13 2012

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A213370.

f:= proc(n) option remember;
procname(n-1) + procname(Bits:-And(2*n,n))
end proc:
f(0):= 1:
seq(f(i),i=0..100); # Robert Israel, Dec 29 2016

A215488[n_] := A215488[n] = If[n == 0, 1, A215488[n-1] + A215488[BitAnd[2*n, n]]];
Array[A215488, 50, 0] (* Paolo Xausa, Aug 06 2025 *)

a = [1]*1000
for n in range(1,777):
    print(a[n-1], end=', ')
    a[n]= a[n-1] + a[2*n & n]

A229762 a(n) = (n XOR floor(n/2)) AND floor(n/2), where AND and XOR are bitwise logical operators.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 1, 0, 4, 4, 5, 4, 2, 2, 1, 0, 8, 8, 9, 8, 10, 10, 9, 8, 4, 4, 5, 4, 2, 2, 1, 0, 16, 16, 17, 16, 18, 18, 17, 16, 20, 20, 21, 20, 18, 18, 17, 16, 8, 8, 9, 8, 10, 10, 9, 8, 4, 4, 5, 4, 2, 2, 1, 0, 32, 32, 33, 32, 34, 34, 33, 32, 36, 36, 37, 36, 34, 34, 33

Offset: 0

Alex Ratushnyak, Sep 28 2013

a(n) has a 01 bit pair in place of each 10 bit pair in n, and everywhere else 0 bits. Or equivalently a(n) has a 1-bit immediately below each run of 1's in n, but excluding a run ending at the least significant bit since below that is below the radix point. - Kevin Ryde, Feb 27 2021

			From _Kevin Ryde_, Feb 27 2021: (Start)
     n = 7267 = binary 1110001100011
  a(n) =  528 = binary   01000010000   1-bit below each run
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191

Cf. A003188 (n XOR floor(n/2)).
Cf. A048724 (n XOR (n*2)).
Cf. A048735 (n AND floor(n/2)).
Cf. A213370 (n AND (n*2)).
Cf. A213064 (n XOR (n*2) AND (n*2), 1-bit above each run).
Cf. A229763 ((2*n) XOR n AND n, low 1-bit each run).

import Data.Bits ((.&.), xor, shiftR)
a229762 n = (n `xor` shiftR n 1) .&. shiftR n 1 :: Int
-- Reinhard Zumkeller, Oct 10 2013

a(n) = bitnegimply(n>>1,n); \\ Kevin Ryde, Feb 27 2021

for n in range(333): print (n ^ (n>>1)) & (n>>1),

def A229762(n): return ~n& n>>1 # Chai Wah Wu, Jun 29 2022

a(n) = (n XOR floor(n/2)) AND floor(n/2) = (n AND floor(n/2)) XOR floor(n/2).

a(n) = floor(n/2) AND NOT n. - Chai Wah Wu, Jun 29 2022

A229763 a(n) = (2*n) XOR n AND n, where AND and XOR are bitwise logical operators.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 1, 8, 9, 10, 9, 4, 5, 2, 1, 16, 17, 18, 17, 20, 21, 18, 17, 8, 9, 10, 9, 4, 5, 2, 1, 32, 33, 34, 33, 36, 37, 34, 33, 40, 41, 42, 41, 36, 37, 34, 33, 16, 17, 18, 17, 20, 21, 18, 17, 8, 9, 10, 9, 4, 5, 2, 1, 64, 65, 66, 65, 68, 69, 66, 65, 72, 73, 74

Offset: 0

Alex Ratushnyak, Sep 28 2013

a(n) is the least significant 1-bit of each run of consecutive 1's in n, and everywhere else 0's. Or equivalently, clear to 0 each 1-bit which has another 1 immediately below. - Kevin Ryde, Feb 27 2021

			From _Kevin Ryde_, Feb 27 2021: (Start)
     n = 1831 = binary 11100100111
  a(n) =  289 = binary   100100001   low 1-bit each run
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191
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, p. 41.

Cf. A003188 (n XOR floor(n/2)).
Cf. A048724 (n XOR (n*2)).
Cf. A048735 (n AND floor(n/2)).
Cf. A213370 (n AND (n*2)).
Cf. A213064 (n XOR (n*2) AND (n*2)).
Cf. A229762 (n XOR floor(n/2) AND floor(n/2), 1-bit below each run).
Cf. A292272 (high 1-bit each run).

import Data.Bits ((.&.), xor, shiftL)
a229763 n = (shiftL n 1 `xor` n) .&. n :: Int
-- Reinhard Zumkeller, Oct 10 2013

Array[BitAnd[BitXor[2 #, #], #] &, 75, 0] (* Michael De Vlieger, Nov 03 2022 *)

a(n) = bitnegimply(n,n<<1); \\ Kevin Ryde, Feb 27 2021

for n in range(333): print (2*n ^ n) & n,

def A229763(n): return n&~(n<<1) # Chai Wah Wu, Jun 29 2022

a(n) = ((2*n) XOR n) AND n = ((2*n) AND n) XOR n.
a(2n) = 2a(n), a(2n+1) = A229762(n). - Ralf Stephan, Oct 07 2013
a(n) = n AND NOT 2n. - Chai Wah Wu, Jun 29 2022
G.f.: x/(1 - x^2) + Sum_{k>=1}(2^k*x^(2^k)/((1 - x)*(1 + x^(2^k))*(1 + x^(2^(k - 1))))). - Miles Wilson, Jan 24 2025

A334076 a(n) = bitwise NOR of n and 2n.

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 1, 0, 7, 4, 1, 0, 3, 0, 1, 0, 15, 12, 9, 8, 3, 0, 1, 0, 7, 4, 1, 0, 3, 0, 1, 0, 31, 28, 25, 24, 19, 16, 17, 16, 7, 4, 1, 0, 3, 0, 1, 0, 15, 12, 9, 8, 3, 0, 1, 0, 7, 4, 1, 0, 3, 0, 1, 0, 63, 60, 57, 56, 51, 48, 49, 48, 39, 36, 33, 32, 35, 32, 33

Offset: 0

Alois P. Heinz, Apr 13 2020

Exactly all bits that are 0 in both parameters (but not a leading 0 of both) are set to 1 in the output of bitwise NOR.

Alois P. Heinz, Table of n, a(n) for n = 0..32767
Wikipedia, Bitwise operation

Cf. A000079, A048724, A125835, A141032, A163617, A213370, A247648.

a:= n-> Bits[Nor](n, 2*n):
seq(a(n), n=0..127);

a(n) = my(x=bitor(n, 2*n)); bitneg(x, #binary(x)); \\ Michel Marcus, Apr 14 2020

def A334076(n):
    m = n|(2*n)
    return 0 if n == 0 else 2**(len(bin(m))-2)-1-m # Chai Wah Wu, Apr 14 2020

a(n) = 0 <=> n in { A247648 } union { 0 }.
a(n) = n-1 <=> n in { A000079 }.
a(n) = n/2 <=> n in { A125835 }.
a(n) = n*3/4 <=> n in { A141032 }.

A344180 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 0, where f(n) = 0 if n is a Fibbinary number (A003714), otherwise f(n) = n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 1, 5, 6, 7, 8, 9, 1, 1, 1, 10, 1, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1, 1, 1, 21, 1, 1, 22, 23, 1, 1, 1, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 1, 1, 1, 45, 1, 1, 46, 47, 1, 1, 1, 48, 49, 50, 51, 52, 1, 1, 1, 53, 1, 1, 54, 55, 56, 57

Offset: 0

Antti Karttunen, May 16 2021

nonn

For all i, j:
  a(i) = a(j) => A085357(i) = A085357(j),
  a(i) = a(j) => A213370(i) = A213370(j),
  a(i) = a(j) => A344182(i) = A344182(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A003714 (positions of 1's), A085357, A213370, A344182.

Cf. also A324400.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux344180(n) = if(!bitand(n,n+n),0,n);
v344180 = rgs_transform(vector(1+up_to,n,Aux344180(n-1)));
A344180(n) = v344180[1+n];

A387269 Numbers whose binary expansion consists of alternating runs of 1's and 0's where each run of 0's is exactly one longer than the preceding run of 1's, and the expansion ends with a 0-run.

Original entry on oeis.org

4, 24, 36, 112, 152, 196, 292, 480, 624, 792, 900, 1176, 1220, 1572, 1984, 2340, 2528, 3184, 3608, 3844, 4720, 4888, 4996, 6296, 6340, 7204, 8064, 9368, 9412, 9764, 10176, 12580, 12768, 14448, 15384, 15876, 18724, 18912, 19568, 19992, 20228, 25200, 25368, 25476

Offset: 1

Ahmet Caglar Saygili, Aug 24 2025

Every term is even (since the binary ends with a 0-run).
Writing the binary word as consecutive pairs 1^{a_1}0^{a_1+1}1^{a_2}0^{a_2+1} ..., the total number of 0's exceeds the total number of 1's by the number of pairs.
Single-pair subfamily 1^{k}0^{k+1}_2, with k >= 1, is A059153(k-1).
Let f(n) = n & (2n) (bitwise AND; A213370). If x has binary run pairs 1^{L_i+1}0^{L_i} (MSB->LSB) as in A387270, then f(x) has pairs 1^{L_i}0^{L_i+1}. Thus f maps A387270 bijectively onto this sequence. Reason: n & (2n) has 1-bits exactly where n has adjacent 1-bits, which shortens every 1-run by 1 and lengthens the following 0-run by 1. Conversely, g(n) = n | floor(n/2) increases each 1-run by 1 and shortens the following 0-run by 1, mapping this sequence back to A387270. Examples: f(6) = 4 (110_2 -> 100_2), f(28) = 24 (11100_2 -> 11000_2); g(24) = 28 (11000_2 -> 11100_2), g(36) = 54 (100100_2 -> 110110_2).

			36 = 100100_2 is a term since its pairs of (1 then 0) runs are (1,2), (1,2).

Ahmet Caglar Saygili, Table of n, a(n) for n = 1..10000
Sean A. Irvine, Java program (github)

Cf. A059153, A166751, A387270, A213370.

function ok(n::Integer)::Bool
    n > 0 && iseven(n) || return false
    x = unsigned(n)
    while x != 0
        z = trailing_zeros(x); x >>= z
        o = trailing_ones(x)
        z == o + 1 || return false
        x >>= o
    end
    true
end
[k for k in 0:10^5 if ok(k)]

isok(k) = if (!(k%2), my(b=binary(k), pos=1, d, dd); for (i=1, #b-1, if (b[i] != b[i+1], if (b[i], d = i-pos+1; pos = i+1, dd = i-pos+1; pos = i+1; if (dd != d+1, return(0))))); dd = #b - pos+1; if (dd != d+1, return(0)); return(1);); \\ Michel Marcus, Aug 26 2025

from itertools import groupby
def ok(n):
    L = [len(list(g)) for k, g in groupby(bin(n)[2:])]
    return (m:=len(L))&1 == 0 and all(L[2*j]+1 == L[2*j+1] for j in range(m>>1))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Aug 25 2025

cp's OEIS Frontend

A163617 a(2*n) = 2*a(n), a(2*n + 1) = 2*a(n) + 2 + (-1)^n, for all n in Z.

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A213541 a(n) = n AND n^2, where AND is the bitwise AND operator.

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A292373 A binary encoding of 3-digits in base-4 representation of n.

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A213540 Numbers k such that k AND k*2 = 2, where AND is the bitwise AND operator.

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A215488 a(0)=1, a(n) = a(n-1) + a(2*n AND n), where AND is the bitwise AND operator.

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A229762 a(n) = (n XOR floor(n/2)) AND floor(n/2), where AND and XOR are bitwise logical operators.

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A229763 a(n) = (2*n) XOR n AND n, where AND and XOR are bitwise logical operators.

A163617 a(2n) = 2a(n), a(2n + 1) = 2a(n) + 2 + (-1)^n, for all n in Z.