A007088 -id:A007088

A099821

Odd positive integers in base 2 (bisection of A007088).

+20
7

1, 11, 101, 111, 1001, 1011, 1101, 1111, 10001, 10011, 10101, 10111, 11001, 11011, 11101, 11111, 100001, 100011, 100101, 100111, 101001, 101011, 101101, 101111, 110001, 110011, 110101, 110111, 111001, 111011, 111101, 111111, 1000001

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

List of binary vectors that begin and end with 1.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Index entries for 2-automatic sequences.

MAPLE

seq(convert(2*n-1, binary), n=1..37); # Emeric Deutsch, Dec 15 2004

MATHEMATICA

FromDigits/@IntegerDigits[Range[1, 71, 2], 2] (* Harvey P. Dale, Apr 05 2019 *)

PROG

(PARI) a(n) = fromdigits(binary(2*n-1), 10); \\ Michel Marcus, Apr 23 2020

CROSSREFS

Cf. A007088, A099820.

KEYWORD

nonn,base,easy

AUTHOR

N. J. A. Sloane, Nov 19 2004

EXTENSIONS

More terms from Emeric Deutsch, Dec 15 2004

STATUS

approved

A122230

a(n) = A007088(A122229(n)).

+20
7

0, 10, 1100, 111000, 11100100, 1110011000, 111001100100, 11100110011000, 1110011001100100, 111001100110011000, 11100110011001100100, 1110011001100110011000, 111001100110011001100100

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..12.

CROSSREFS

Cf. A080070, A122233, A122236, A122240, A122243, A122246.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Sep 14 2006

STATUS

approved

A122233

a(n) = A007088(A122232(n)).

+20
7

101010, 11010100, 1111100000, 111100100100, 11110001011000, 1111110010000100, 111101100010011000, 11110001101100100100, 1111101011000110000100, 111101100110001100010100

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..10.

CROSSREFS

Cf. A080070, A122230, A122236, A122240, A122243, A122246.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Sep 14 2006

STATUS

approved

A122243

a(n) = A007088(A122242(n)).

+20
7

101010, 11110000, 1110010100, 111010100100, 11101001110000, 1111001010011000, 111100100101110000, 11101100101001011000, 1110110010011011100000, 111100111001000110110000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..10.

CROSSREFS

A080070, A122230, A122233, A122236, A122240, A122246.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Sep 14 2006

STATUS

approved

A122246

a(n) = A007088(A122245(n)).

+20
7

101100, 11101000, 1110011000, 111100100100, 11101001011000, 1110110010011000, 111100100110110000, 11110011010001011000, 1110110010010110101000, 111011001110000110110000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..10.

CROSSREFS

A080070, A122230, A122233, A122236, A122240, A122243.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Sep 14 2006

STATUS

approved

A174203

In the sequence A007088 of binary representations of n, delete the 1st, 3rd, 5th, 7th, 9th etc occurrence of 0 and 1. Rewrite the bits left over in base 10.

+20
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Deleting the odd occurrences of 0 and the even occurrences of 1, we get the same sequence apart from the initial 2 replaced by 1 and 0.

LINKS

Table of n, a(n) for n=0..78.

EXAMPLE

Numbers in base 2: 0, 1, 10, 11, 100, 101, 110, 111, 1000, .... Delete odd occurrences of 0 (replaced by ~): ~, 1, 10, 11, 1~0, 1~1, 110, 111, 1~0~ Delete odd occurrences of 1 (replaced by ~): ~, ~, 10, ~1, ~~0, 1~~, 1~0, 1~1, ~~0~.

The pieces left are 10, 1, 0, 1, 10, 11, 0, ... which is in base 10: 2, 1, 0, 1, 2, 3, 0, ....

MAPLE

n1 := 1 ; n0 := 1 ;

for n from 0 to 80 do

if n = 0 then b := [0] ; else b := convert(n, base, 2) ; end if;

for d from nops(b) to 1 by -1 do

if op(d, b) = 0 then

if type(n0, 'odd') then b := subsop(d=NULL, b) ; end if; n0 := n0+1 ;

else

if type(n1, 'odd') then b := subsop(d=NULL, b) ; end if; n1 := n1+1 ;

end if;

end do:

add(2^(i-1)*op(i, b), i=1..nops(b)) ; printf("%d, ", %) ;

end do: # R. J. Mathar, Nov 23 2010

CROSSREFS

Cf. A174204, A174205, A174206, A174207, A174208, A174209, A174210.

KEYWORD

easy,nonn,base,less

AUTHOR

Paolo P. Lava and Giorgio Balzarotti, Mar 15 2010

EXTENSIONS

Sequence extended, keyword:base,less added by R. J. Mathar, Nov 23 2010

STATUS

approved

A122236

a(n) = A007088(A122235(n)).

+20
6

101100, 11011000, 1111001000, 111100010100, 11111100000100, 1111001001110000, 111100010111000100, 11111111000010000100, 1111100010011011000100, 111101101001110000100100

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..10.

CROSSREFS

Cf. A080070, A122230, A122233, A122240, A122243, A122246.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Sep 14 2006

STATUS

approved

A122240

a(n) = A007088(A122239(n)).

+20
6

110100, 11110000, 1111000100, 111110000100, 11110011000100, 1111110000010100, 111100100111100000, 11110001011100100100, 1111111000101100000100, 111100101001101001110000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..10.

CROSSREFS

Cf. A080070, A122230, A122233, A122236, A122243, A122246.

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Sep 14 2006

STATUS

approved

A080317

a(n) = A007088(A080316(n)).

+20
3

0, 1100, 11001100, 11110000, 110011001100, 110011110000, 111111000000, 111100001100, 111100110000, 1100110011001100, 1100110011110000, 1100111111000000, 1100111100001100, 1100111100110000, 1111110000001100

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Note that here we have totally balanced binary sequences (as in A014486/A063171), 10, 1010, 1100, 101010, 101100, 110010, ..., but with each bit duplicated. Furthermore, the order is a bit different.

LINKS

Table of n, a(n) for n=0..14.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 02 2003

STATUS

approved

A099820

Even nonnegative integers in base 2 (bisection of A007088).

+20
3

0, 10, 100, 110, 1000, 1010, 1100, 1110, 10000, 10010, 10100, 10110, 11000, 11010, 11100, 11110, 100000, 100010, 100100, 100110, 101000, 101010, 101100, 101110, 110000, 110010, 110100, 110110, 111000, 111010, 111100, 111110, 1000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..32.

Index entries for 2-automatic sequences.

MAPLE

seq(convert(2*n, binary), n=0..37); # Emeric Deutsch, Dec 15 2004

CROSSREFS

Cf. A007088, A099821.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 19 2004

EXTENSIONS

More terms from Emeric Deutsch, Dec 15 2004

STATUS

approved