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A025480 a(2n) = n, a(2n+1) = a(n). 44
0, 0, 1, 0, 2, 1, 3, 0, 4, 2, 5, 1, 6, 3, 7, 0, 8, 4, 9, 2, 10, 5, 11, 1, 12, 6, 13, 3, 14, 7, 15, 0, 16, 8, 17, 4, 18, 9, 19, 2, 20, 10, 21, 5, 22, 11, 23, 1, 24, 12, 25, 6, 26, 13, 27, 3, 28, 14, 29, 7, 30, 15, 31, 0, 32, 16, 33, 8, 34, 17, 35, 4, 36, 18, 37, 9, 38, 19, 39, 2, 40, 20, 41, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

These are the Grundy values or nim-values for heaps of n beans in the game where you're allowed to take up to half of the beans in a heap. - R. K. Guy, Mar 30 2006. See Levine 2004/2006 for more about this. - N. J. A. Sloane, Aug 14 2016

When n>0 is written as (2k+1)*2^j then k=A000265(n-1) and j=A007814(n), so: when n is written as (2k+1)*2^j-1 then k=A025480(n) and j=A007814(n+1), when n>1 is written as (2k+1)*2^j+1 then k=A025480(n-2) and j=A007814(n-1). - Henry Bottomley, Mar 02 2000

According to the comment from Deuard Worthen (see Example section), this may be regarded as a triangle where row r=1,2,3... has length 2^(r-1) and values T[r,2k-1]=T[r-1,k], T[r,2k]=2^(r-1)+k-1, i.e., previous row gives first, 3rd, 5th, ... term and 2nd, 4th, ... terms are numbers 2^(r-1),...,2^r-1 (i.e., those following the last one from the previous row). - M. F. Hasler, May 03 2008

Let StB be a Stern-Brocot tree hanging between (pseudo)fractions Left and Right, then StB(1) = mediant(Left,Right) and for n>1: StB(n) = if a(n-1)<>0 and a(n)<>0 then mediant(StB(a(n-1)),StB(a(n))) else if a(n)=0 then mediant(StB(a(n-1)),Right) else mediant(Left,StB(a(n-1))), where mediant(q1,q2) = ((numerator(q1)+numerator(q2)) / (denominator(q1)+denominator(q2))). - Reinhard Zumkeller, Dec 22 2008

This sequence is the unique fixed point of the function (a(0), a(1), a(2), ...) |--> (0, a(0), 1, a(1), 2, a(2), ...) which interleaves the nonnegative integers between the elements of a sequence. - Cale Gibbard (cgibbard(AT)gmail.com), Nov 18 2009

The following relation holds: 2^A007814(n+1)*(2*A025480(n)+1)=A001477(n+1). (See functions hd,tl and cons in [Paul Tarau 2009].) - Paul Tarau (paul.tarau(AT)gmail.com), Mar 21 2010

Also the number of remaining survivors in a Josephus problem after the person originally first in line has been eliminated (see A225381). - Marcus Hedbring, May 18 2013

A fractal sequence - see Levine 2004/2006. - N. J. A. Sloane, Aug 14 2016

From David James Sycamore, Apr 29 2020 : (Start)

One of a family of fractal sequences, S_k; defined as follows for k>=2: a(k*n) = n, a(k*n+r) = a((k-1)*n + (r-1)), r = 1..(k-1). S_2 is A025480; S_3 gives: a(3*n) = n,  a(3*n + 1) = a(2*n), a(3*n + 2) = a(2*n + 1), which is A263390 (whose name is not expressed entirely in this form, but it is the same sequence).

Sequences corresponding to higher k values do not appear to be recorded in oeis. The subsequence of all non zero terms is A131987. Thus A025480 has a proper subsequence identical to itself and another proper subsequence different from itself, which is also fractal (End).

Similar to but different from A108202. - N. J. A. Sloane, Nov 26 2020

REFERENCES

L. Levine, Fractal sequences and restricted Nim, Ars Combin. 80 (2006), 113-127.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Josef Eschgfäller, Andrea Scarpante, Dichotomic random number generators, arXiv:1603.08500 [math.CO], 2016.

R. Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 3.2.4.

L. Levine, Fractal sequences and restricted Nim, arXiv:math/0409408 [math.CO], 2004.

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Paul Tarau, A Groupoid of Isomorphic Data Transformations, Calculemus 2009, 8th International Conference, MKM 2009, pp. 170-185, Springer, LNAI 5625.

Index entries for sequences related to the Josephus Problem

FORMULA

a(n) = (A000265(n+1)-1)/2 = ((n+1)/A006519(n+1)-1)/2.

a(n) = A153733(n)/2. - Reinhard Zumkeller, Dec 31 2008

a(3*n + 1) = A173732(n). - Reinhard Zumkeller, Apr 29 2012

a((2*n+1)*2^p-1) = n, p >= 0 and n >= 0. - Johannes W. Meijer, Jan 24 2013

a(n) = n - A225381(n). - Marcus Hedbring, May 18 2013

G.f.: -1/(1-x) + Sum_{k>=0} x^(2^k-1)/(1-2*x^2^(k+1)+x^2^(k+2)). - Ralf Stephan, May 19 2013

a(n) = A049084(A181363(n+1)). - Reinhard Zumkeller, Mar 22 2014

a(n) = floor( n/ 2^A001511(n+1) ). - Adam Shelly, Mar 05 2019

Recursion: a(0) = 0; a(n + 1) = a(a(n)) if a(n) is a first occurrence of a term, else a(n + 1) = n - a(n-1). - David James Sycamore, Apr 29 2020

EXAMPLE

From Deuard Worthen (deuard(AT)raytheon.com), Jan 27 2006: (Start)

The sequence can be constructed as a triangle as:

  0

  0  1

  0  2  1  3

  0  4  2  5  1  6  3  7

  0  8  4  9  2 10  5 11  1 12  6 13  3 14  7 15

  ...

At each stage we interleave the next 2^m numbers in the previous row. (End)

Left=0/1, Right=1/0: StB=A007305/A047679; Left=0/1, Right=1/1: StB=A007305/A007306; Left=1/3, Right=2/3: StB=A153161/A153162. - Reinhard Zumkeller, Dec 22 2008

MAPLE

a:=array[0..10001]; M:=5000; for n from 0 to M do a[2*n]:=n; a[2*n+1]:=a[n]; od: for n from 0 to 2*M do lprint(n, a[n]); od:

nmax := 83: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := n od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 24 2013

A025480 := proc(n)

    option remember ;

    if type(n, 'even') then

        n/2 ;

    else

        procname((n-1)/2) ;

    end if;

end proc: # R. J. Mathar, Jul 16 2020

MATHEMATICA

a[n_] := a[n] = If[OddQ@n, a[(n - 1)/2], n/2]; Table[ a[n], {n, 0, 83}] (* Robert G. Wilson v, Mar 30 2006 *)

Table[BitShiftRight[n, IntegerExponent[n, 2] + 1], {n, 100}] (* IWABUCHI Yu(u)ki, Oct 13 2012 *)

PROG

(PARI) a(n)={while(n%2, n\=2); n\2} \\ M. F. Hasler, May 03 2008

(PARI) A025480(n)=n>>valuation(n*2+2, 2) \\ M. F. Hasler, Apr 12 2012

(Haskell)

import Data.List

interleave xs ys = concat . transpose $ [xs, ys]

a025480 = interleave [0..] a025480

-- _Cale Gibbard_, Nov 18 2009:

(Haskell)  Cf. comments by Worthen and Hasler.

import Data.List (transpose)

a025480 n k = a025480_tabf !! n !! k

a025480_row n = a025480_tabf !! n

a025480_tabf = iterate (\xs -> concat $

   transpose [xs, [length xs .. 2 * length xs - 1]]) [0]

a025480_list = concat $ a025480_tabf

-- Reinhard Zumkeller, Apr 29 2012

(Sage)

A025480 = lambda n: odd_part(n+1)//2

[A025480(n) for n in (0..83)] # Peter Luschny, May 20 2014

CROSSREFS

a(n) = A003602(n)-1.

The Y-projection of A075300.

Cf. A108202, A138002, A000265, A003602, A103391, A153733, A220466, A225381, A131987, A263390.

Sequence in context: A334594 A062778 A108202 * A088002 A030109 A208571

Adjacent sequences:  A025477 A025478 A025479 * A025481 A025482 A025483

KEYWORD

easy,nonn,nice,tabf,hear,changed

AUTHOR

David W. Wilson

EXTENSIONS

Edited by M. F. Hasler, Mar 16 2018

STATUS

approved

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Last modified November 29 19:26 EST 2020. Contains 338769 sequences. (Running on oeis4.)