A174208 -id:A174208

Search: a174208 -id:a174208

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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.

+10
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

A174207

Depleted stream of the natural numbers written in base 2: delete even occurrences of digit 0 and odd occurrences of digit 1.

+10
4

0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0

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

OFFSET

0,1

COMMENTS

The numbers in base 2: 0, 1, 10, 11, 100, 101, 110, 111, 1000, ....

The infinite stream of digits: 0110111001011101111000...

Delete even (2nd, 4th, 6th, ...) occurrences of 0 (replaced by ~):

011~1110~10111~11110~0...

Delete odd (1st, 3rd, 5th, ...) occurrences of 1 (replaced by ~):

0~1~~1~0~10~1~~1~1~0~0...

Digits remaining define the sequence: 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, ....

LINKS

Table of n, a(n) for n=0..118.

MAPLE

nb := 100 ;

L := [0] ;

for n from 1 to nb do

nn := ListTools[Reverse](convert(n, base, 2)) ;

L := [op(L), op(nn)] ;

end do;

Lout := [] ;

ie :=1 ;

io :=1 ;

for i from 1 to nops(L) do

if op(i, L) =0 then

if ie>0 then

if type(ie, 'odd') then

printf("%d, ", 0) ;

end if;

ie := ie+1 ;

else

if io>0 then

if type(io, 'even') then

printf("%d, ", 1) ;

end if;

io := io+1 ;

end if;

end do: # R. J. Mathar, Dec 06 2011

MATHEMATICA

nMax = 60; bits = Join @@ Table[IntegerDigits[n, 2], {n, 0, nMax}]; pos0 = Position[bits, 0] // Flatten; even0 = Partition[pos0, 2][[All, 2]]; bits2 = ReplacePart[bits, Alternatives @@ even0 -> "~"]; pos1 = Position[bits2, 1] // Flatten; odd1 = Partition[pos1, 2][[All, 1]]; DeleteCases[ ReplacePart[ bits2, Alternatives @@ odd1 -> "~"], "~"] (* Jean-François Alcover, Nov 18 2016 *)

CROSSREFS

Cf. A174203, A174204 (base 10), A174205, A174206, A174208-A174210.

KEYWORD

easy,nonn,base

AUTHOR

Paolo P. Lava and Giorgio Balzarotti, Mar 15 2010

STATUS

approved

A174209

In the sequence of natural numbers, moving left to right, delete 1st, 3rd, 5th, 7th etc occurrence of each of the ten digits.

+10
3

10, 1, 2, 13, 4, 15, 6, 17, 8, 19, 2, 2, 2, 2, 2, 30, 1, 32, 3, 4, 35, 6, 37, 8, 39, 4, 4, 4, 4, 4, 50, 1, 52, 3, 54, 5, 6, 57, 8, 59, 6, 6, 6, 6, 6, 70, 1, 72, 3, 74, 5, 76, 7, 8, 79, 8, 8, 8, 8, 8, 90, 1, 92, 3, 94, 5, 96, 7, 98, 9, 0, 10, 1, 0, 1, 0, 1, 0, 1, 0, 1, 11, 12, 13, 14, 15, 16, 17

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

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..87.

EXAMPLE

Starting from

0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,1,20,21,22,23,24,25,26,27,28,29,..

Delete odd occurrences of digit 0 (replaced by ~):

,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,2~,21,22,23,24,25,26,27,28,29,..

Delete odd occurrences of digit 1 (replaced by ~):

,~,2,3,4,5,6,7,8,9,10,~1,~2,13,~4,15,~6,17,~8,19,2~,2~,22,23,24,25,26,27,28,29 ,..

Delete odd occurrences of digit 2 (replaced by ~):

,~,~,3,4,5,6,7,8,9,10,~1,~2,13,~4,15,~6,17,~8,19,~~,2~,~2,~3,24,~5,26,~7,28,~9 ,..

Delete odd occurrences of digit 3 (replaced by ~):

,~,~,~,4,5,6,7,8,9,10,~1,~2,13,~4,15,~6,17,~8,19,~~,2~,~2,~~,24,~5,26,~7,28,~9,..

Delete odd occurrences of digit 4 (replaced by ~):

,~,~,~,~,5,6,7,8,9,10,~1,~2,13,~4,15,~6,17,~8,19,~~,2~,~2,~~,2~,~5,26,~7,28,~9,..

Delete odd occurrences of digit 5 (replaced by ~):

,~,~,~,~,~,6,7,8,9,10,~1,~2,13,~4,15,~6,17,~8,19,~~,2~,~2,~~,2~,~~,26,~7,28,~9,..

Delete odd occurrences of digit 6 (replaced by ~):

,~,~,~,~,~,~,7,8,9,10,~1,~2,13,~4,15,~6,17,~8,19,~~,2~,~2,~~,2~,~~,2~,~7,28,~9,..

Delete odd occurrences of digit 7 (replaced by ~):

,~,~,~,~,~,~,~,8,9,10,~1,~2,13,~4,15,~6,17,~8,19,~~,2~,~2,~~,2~,~~,2~,~~,28,~9,..

Delete odd occurrences of digit 8 (replaced by ~):

,~,~,~,~,~,~,~,~,9,10,~1,~2,13,~4,15,~6,17,~8,19,~~,2~,~2,~~,2~,~~,2~,~~,2~,~9,..

Delete odd occurrences of digit 9 (replaced by ~):

,~,~,~,~,~,~,~,~,~,10,~1,~2,13,~4,15,~6,17,~8,19,~~,2~,~2,~~,2~,~~,2~,~~,2~,~~,..

The residual digits define the sequence: 10, 1, 2, 13, 4, 15, 6, 17, 8, 19, 2, 2, 2, 2, 2,....

CROSSREFS

Cf. A174203, A174204, A174205, A174206, A174207, A174208, A174210.

KEYWORD

easy,nonn,base

AUTHOR

Paolo P. Lava & Giorgio Balzarotti, Mar 15 2010

STATUS

approved

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