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A025480 a(2n) = n, a(2n+1) = a(n). 33
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 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

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)

According to the comment from Deuard Worthen, 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]

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

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

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

REFERENCES

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

LINKS

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

L. Levine, Fractal sequences and restricted Nim

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.

FORMULA

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

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

EXAMPLE

Comment from Deuard Worthen (deuard(AT)raytheon.com), Jan 27 2006: This sequence can be constructed as a triangle, thus:

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 interpolate the next 2^m numbers in the previous row.

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

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) A025480(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

The Y-projection of A075300.

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

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

Sequence in context: A081171 A062778 A108202 * A088002 A030109 A208571

Adjacent sequences:  A025477 A025478 A025479 * A025481 A025482 A025483

KEYWORD

easy,nonn,nice,tabf,hear

AUTHOR

David W. Wilson

EXTENSIONS

Additional comments from Henry Bottomley, Mar 02 2000

STATUS

approved

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Last modified November 23 22:48 EST 2014. Contains 249866 sequences.