cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A059009 Numbers having an odd number of zeros in their binary expansion.

Original entry on oeis.org

0, 2, 5, 6, 8, 11, 13, 14, 17, 18, 20, 23, 24, 27, 29, 30, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61, 62, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 123, 125, 126, 128, 131
Offset: 0

Views

Author

Patrick De Geest, Dec 15 2000

Keywords

Comments

Positions of ones in A059448 for n >= 1. - John Keith, Mar 09 2022

Examples

			18 is in the sequence because 18 = 10010_2. '10010' has three zeros. - _Indranil Ghosh_, Feb 04 2017

Links

Crossrefs

Cf. A000069, A001969, A059010, A059011, A059012, A059013, A059014, A059448.
Cf. A023416.

Programs

  • Haskell
    a059009 n = a059009_list !! (n-1)
a059009_list = filter (odd . a023416) [1..]
-- Reinhard Zumkeller, Jan 21 2014
  • Maple
    a:= proc(n) option remember;
  if n::even then -a(n/2) + 3*n + 1 else a((n-1)/2) + n + 1 fi
end proc:
a(0):= 0:
seq(a(n),n=0..100); # Robert Israel, Feb 23 2016
  • Mathematica
    Select[Range[0,150],OddQ[Count[IntegerDigits[#,2],0]]&] (* Harvey P. Dale, Oct 22 2011 *)
  • PARI
    is(n)=hammingweight(bitneg(n,#binary(n)))%2 \\ Charles R Greathouse IV, Mar 26 2013
  • PARI
    a(n) = if(n==0,0, 2*n + (logint(n,2) - hammingweight(n) + 1) % 2); \\ Kevin Ryde, Mar 11 2021
  • Python
    i=j=0
while j<=800:
    if bin(i)[2:].count("0")%2:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 04 2017
  • R
    maxrow <- 4 # by choice
onezeros <- 1
for(m in 1:(maxrow+1)){
  row <- onezeros[2^(m-1):(2^m-1)]
  onezeros <- c(onezeros, c(1-row, row) )
}
a <- which(onezeros == 0)
a
# Yosu Yurramendi, Mar 28 2017

Formula

a(0) = 0, a(2*n) = -a(n) + 6*n + 1, a(2*n+1) = a(n) + 2*n + 2. a(n) = 2*n + 1/2(1-(-1)^A023416(n)) = 2*n + A059448(n). - Ralf Stephan, Sep 17 2003

A059010 Natural numbers having an even number of nonleading zeros in their binary expansion.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 12, 15, 16, 19, 21, 22, 25, 26, 28, 31, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 63, 64, 67, 69, 70, 73, 74, 76, 79, 81, 82, 84, 87, 88, 91, 93, 94, 97, 98, 100, 103, 104, 107, 109, 110, 112, 115, 117, 118, 121, 122, 124, 127, 129, 130
Offset: 0

Views

Author

Patrick De Geest, Dec 15 2000

Keywords

Comments

Positions of ones in A298952, and of zeros in A059448. - John Keith, Mar 09 2022

Links

Crossrefs

Cf. A059009 (complement).
Cf. A023416 (number of 0-bits), A059448 (their parity), A298952 (opposite parity).
Cf. A001969 (even 1-bits), A059012 (even both 0's and 1's), A059014 (even 0's, odd 1's).

Programs

  • Haskell
    a059010 n = a059010_list !! (n-1)
a059010_list = filter (even . a023416) [1..]
-- Reinhard Zumkeller, Jan 21 2014
  • Mathematica
    Select[Range[130], EvenQ @ DigitCount[#, 2, 0] &] (* Jean-François Alcover, Apr 11 2011 *)
  • PARI
    is(n)=hammingweight(bitneg(n,#binary(n)))%2==0 \\ Charles R Greathouse IV, Mar 26 2013
  • PARI
    a(n) = if(n==0,1, 2*n + (logint(n,2) - hammingweight(n)) % 2); \\ Kevin Ryde, Mar 11 2021
  • Python
    #Program to generate the b-file
i=1
j=0
while j<=250:
    if bin(i)[2:].count("0")%2==0:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017
  • R
    maxrow <- 4 # by choice
onezeros <- 1
for(m in 1:(maxrow+1)){
  row <- onezeros[2^(m-1):(2^m-1)]
  onezeros <- c(onezeros, c(1-row, row) )
}
a <- which(onezeros == 1)
a
# Yosu Yurramendi, Mar 28 2017

Formula

a(0) = 1, a(2n) = -a(n) + 6n + 1, a(2n+1) = a(n) + 2n + 2. a(n) = 2n+1 - 1/2(1-(-1)^A023416(n)) = 2n+1 - A059448(n). - Ralf Stephan, Sep 17 2003

Extensions

Name clarified by Antti Karttunen, Mar 28 2017

A333441 Numbers where each binary digit can be paired with a digit of the same value at another position so that two pairs can be nested but cannot otherwise overlap.

Original entry on oeis.org

0, 3, 9, 12, 15, 33, 36, 39, 45, 48, 51, 54, 57, 60, 63, 129, 132, 135, 141, 144, 147, 150, 153, 156, 159, 165, 177, 180, 183, 189, 192, 195, 198, 201, 204, 207, 210, 216, 219, 222, 225, 228, 231, 237, 240, 243, 246, 249, 252, 255, 513, 516, 519, 525, 528, 531
Offset: 1

Views

Author

Rémy Sigrist, Mar 21 2020

Keywords

Comments

The term 0 is included by convention (we consider here that it has no digit).
This sequence is a binary variant of A333440.
Every term belong to A059012.
This sequence has connections with A014486; in both sequences digits are balanced in some way.

Examples

			The first terms, alongside their binary representation with a possible pairing, are:
  n   a(n)  bin(a(n))
  --  ----  ------------
   1     0  0
   2     3  (11)
   3     9  (1(00)1)
   4    12  (11)(00)
   5    15  (11)(11)
   6    33  (1(00)(00)1)
   7    36  (1(00)1)(00)
   8    39  (1(00)1)(11)
   9    45  (1(0(11)0)1)
  10    48  (11)(00)(00)
  11    51  (11)(00)(11)
  12    54  (11)(0(11)0)
  13    57  (11)(1(00)1)
  14    60  (11)(11)(00)
  15    63  (11)(11)(11)

Links

Crossrefs

Cf. A014486, A059012, A333440.

Programs

  • PARI
    is(n, base=2) = { my (u=0, s=0); while (n, my (d=n%base); if (u && s%base==d, u--; s\=base, u++; s=s*base+d); n\=base); u==0 }
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.