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A030101 a(n) is the number produced when n is converted to binary digits, the binary digits are reversed and then converted back into a number. 101
0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 13, 3, 11, 7, 15, 1, 17, 9, 25, 5, 21, 13, 29, 3, 19, 11, 27, 7, 23, 15, 31, 1, 33, 17, 49, 9, 41, 25, 57, 5, 37, 21, 53, 13, 45, 29, 61, 3, 35, 19, 51, 11, 43, 27, 59, 7, 39, 23, 55, 15, 47, 31, 63, 1, 65, 33, 97, 17, 81, 49, 113, 9, 73, 41, 105, 25, 89, 57 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

As with decimal reversal, initial zeros are ignored; otherwise, the reverse of 1 would be 1000000... ad infinitum.

Numerators of the binary van der Corput sequence. - Eric Rowland, Feb 12 2008

It seems that in most cases A030101[x]=A000265[x] and that if A030101[x]<>A000265[x], the next time A030101[y]=A000265[x], A030101[x]=A000265[y]. Also, it seems that if a pair of values exist at one index, they will exist at any index where one of them exist. It also seems like the greater of the pair always shows up on A000265 first. - Dylan Hamilton, Aug 04 2010

The number of occasions A030101(n)=A000265(n) before n=2^k is A053599(k)+1. For n=0..2^19, the sequences match less than 1% of the time. - Andrew Woods, May 19 2012

For n > 0: a(a(n)) = n iff n is odd; a(A006995(n)) = A006995(n). - Juli Mallett, Nov 11 2010, corrected: Reinhard Zumkeller, Oct 21 2011

n is binary palindromic iff a(n) = n. - Reinhard Zumkeller, corrected: Jan 17 2012, thanks to Hieronymus Fischer, who pointed this out; Oct 21 2011

Given any n>1, the set of numbers A030109[i]=(A030101[i]-1)/2 for indexes i ranging from 2^n to 2^(n+1)-1 is a permutation of the set of consecutive integers {0,1,2,...,2^n-1}. This is important in the standard FFT algorithms (starting or ending bit-reversal permutation). - Stanislav Sykora, Mar 15 2012

Row n of A030308 gives the binary digits of a(n), prepended with zero at even positions. - Reinhard Zumkeller, Jun 17 2012

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

Michael Gilleland, Some Self-Similar Integer Sequences

Project Euler, Problem 463

Seventeenth annual USA Mathematical Olympiads, Math. Mag., 62 (1989), 210-214 (#3).

Wikipedia, van der Corput sequence.

Index entries for sequences related to binary expansion of n

FORMULA

a(n) = 0, a(2n) = a(n), a(2n+1) = a(n) + 2^([log2(n)]+1). For n>0, a(n) = 2*A030109(n) - 1. - Ralf Stephan, Sep 15 2003

a(n) = b(n,0) with b(n,r) = if n=0 then r else b(floor(n/2), 2*r + n mod 2). - Reinhard Zumkeller, Mar 03 2010

a(1)=1, a(3)=3, a(2n)=a(n), a(4n+1)=2a(2n+1)-a(n), a(4n+3)=3a(2n+1)-2a(n) (as in the Project Euler problem). To prove this, expand the recurrence into binary strings and reversals. - David Applegate, Mar 16 2014, following a posting to the Sequence Fans Mailing List by Martin Møller Skarbiniks Pedersen.

EXAMPLE

a(100) = 19 because 100 (base 10) = 1100100 (base 2) and R(1100100 (base 2)) = 10011 (base 2) = 19 (base 10).

MAPLE

A030101 := proc(n)

    convert(n, base, 2) ;

    ListTools[Reverse](%) ;

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

end proc: # R. J. Mathar, Mar 10 2015

# second Maple program:

a:= proc(n) local m, r; m:=n; r:=0;

      while m>0 do r:=r*2+irem(m, 2, 'm') od; r

    end:

seq(a(n), n=0..80);  # Alois P. Heinz, Nov 17 2015

MATHEMATICA

Table[FromDigits[Reverse[IntegerDigits[i, 2]], 2], {i, 0, 80}]

f[n_] := Switch[Mod[n, 4], 0, f[n/2], 1, 2 f[(n + 1)/2] - f[(n - 1)/4], 2, f[n/2], 3, 3 f[(n - 1)/2] - 2 f[(n - 3)/4]]; f[0] = 0; f[1] = 1; f[3] = 3; Array[f, 80, 0] (* Robert G. Wilson v, Mar 18 2014 *)

PROG

(PARI) a(n)=if(n<1, 0, subst(Polrev(binary(n)), x, 2))

(MAGMA) A030101:=func<n|SequenceToInteger(Reverse(IntegerToSequence(n, 2)), 2)>; // Jason Kimberley, Sep 19 2011

(Haskell)

a030101 = f 0 where

   f y 0 = y

   f y x = f (2 * y + b) x'  where (x', b) = divMod x 2

-- Reinhard Zumkeller, Mar 18 2014, Oct 21 2011

(Sage)

def A030101(n): return Integer(bin(n).lstrip("0b")[::-1], 2) if n<>0 else 0

[A030101(n) for n in (0..78)]  # Peter Luschny, Aug 09 2012

(Python)

def a(n): return int(bin(n)[2:][::-1], 2) # Indranil Ghosh, Apr 24 2017

CROSSREFS

Cf. A030102 - A030109, A036044, A056539, A004086, A005408.

Cf. A055944 (reverse and add), A178225.

Sequence in context: A106609 A171968 A093474 * A162742 A081432 A136655

Adjacent sequences:  A030098 A030099 A030100 * A030102 A030103 A030104

KEYWORD

nonn,base,nice,hear,look

AUTHOR

David W. Wilson

EXTENSIONS

Edits (including correction of an erroneous date pointed out by J. M. Bergot) by Jon E. Schoenfield, Mar 16 2014

STATUS

approved

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Last modified August 18 17:58 EDT 2017. Contains 290732 sequences.