login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A086449 a(0) = 1, a(2n+1) = a(n), a(2n) = a(n) + a(n-1) + ... + a(n-2^m) + ... where a(n) = 0 for n < 0. 4
1, 1, 2, 1, 4, 2, 4, 1, 8, 4, 8, 2, 12, 4, 8, 1, 18, 8, 16, 4, 26, 8, 16, 2, 34, 12, 24, 4, 36, 8, 16, 1, 48, 18, 36, 8, 60, 16, 32, 4, 80, 26, 52, 8, 78, 16, 32, 2, 104, 34, 68, 12, 110, 24, 48, 4, 136, 36, 72, 8, 108, 16, 32, 1, 154, 48, 96, 18, 160, 36, 72, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Conjecture: all a(n) are even except a(2^k-1) = 1. Also a(2^k-2) = 2^(k-1). [For proof see link.]

Setting m=0 gives Stern-Brocot sequence (A002487).

a(n) is the number of ways of writing n as a sum of powers of 2, where each power appears p times, with p itself a power of 2.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Lambert Herrgesell, Proof of conjecture

Peter Luschny, Rational Trees and Binary Partitions.

FORMULA

G.f.: Product_{k>=0} (1 + Sum_{j>=0} x^(2^(k+j)). [Corrected by Herbert S. Wilf, May 31 2006]

EXAMPLE

From Peter Luschny, Sep 01 2019: (Start)

The sequence splits into rows of length 2^k:

1

1,  2

1,  4, 2,  4

1,  8, 4,  8, 2, 12, 4,  8

1, 18, 8, 16, 4, 26, 8, 16, 2, 34, 12, 24, 4, 36, 8, 16

.

The first few partitions counted are (compare with the list in A174980):

[ 0]  [[]]

[ 1]  [[1]]

[ 2]  [[2], [1, 1]]

[ 3]  [[2, 1]]

[ 4]  [[4], [2, 2], [2, 1, 1], [1, 1, 1, 1]]

[ 5]  [[4, 1], [2, 2, 1]]

[ 6]  [[4, 2], [4, 1, 1], [2, 2, 1, 1], [2, 1, 1, 1, 1]]

[ 7]  [[4, 2, 1]]

[ 8]  [[8], [4, 4], [4, 2, 2], [4, 2, 1, 1], [4, 1, 1, 1, 1], [2, 2, 2, 2],

      [2, 2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]]

[ 9]  [[8, 1], [4, 4, 1], [4, 2, 2, 1], [2, 2, 2, 2, 1]]

[10]  [[8, 2], [8, 1, 1], [4, 4, 2], [4, 4, 1, 1], [4, 2, 2, 1, 1],

      [4, 2, 1, 1, 1, 1], [2, 2, 2, 2, 1, 1], [2, 1, 1, 1, 1, 1, 1, 1, 1]]

[11]  [[8, 2, 1], [4, 4, 2, 1]]

[12]  [[8, 4], [8, 2, 2], [8, 2, 1, 1], [8, 1, 1, 1, 1], [4, 4, 2, 2],

      [4, 4, 2, 1, 1], [4, 4, 1, 1, 1, 1], [4, 2, 2, 2, 2], [4, 2, 2, 1, 1, 1, 1],

      [4, 1, 1, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 1, 1, 1, 1],

      [2, 2, 1, 1, 1, 1, 1, 1, 1, 1]]

[13]  [[8, 4, 1], [8, 2, 2, 1], [4, 4, 2, 2, 1], [4, 2, 2, 2, 2, 1]]

[14]  [[8, 4, 2], [8, 4, 1, 1], [8, 2, 2, 1, 1], [8, 2, 1, 1, 1, 1],

      [4, 4, 2, 2, 1, 1], [4, 4, 2, 1, 1, 1, 1], [4, 2, 2, 2, 2, 1, 1],

      [4, 2, 1, 1, 1, 1, 1, 1, 1, 1]]

[15]  [[8, 4, 2, 1]]

(End)

MAPLE

A086449 := proc(n) option remember;

local IndexSet, k; IndexSet := proc(n) local i, j, z;

i := iquo(n, 2); j := i; if odd::n then i := i-1; z := 1;

while 0 <= i do j := j, i; i := i-z; z := z+z od fi; j end:

if n < 2 then 1 else add(A086449(k), k=IndexSet(n)) fi end:

seq(A086449(i), i=0..71); # Peter Luschny, May 06 2011

# second Maple program:

a:= proc(n) option remember; local r; `if`(n=0, 1,

      `if`(irem(n, 2, 'r')=1, a(r),

       a(r) +add(a(r-2^m), m=0..ilog2(r))))

    end:

seq(a(n), n=0..80);  # Alois P. Heinz, May 30 2014

MATHEMATICA

nn=30; CoefficientList[Series[Product[1+Sum[x^(2^(k+j)), {j, 0, nn}], {k, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, May 30 2014 *)

PROG

a(n)=local(k): if(n<1, n>=0, if(n%2==0, a(n/2)+sum(k=0, n, a((n-2^(k+1))/2)), a((n-1)/2)))

(MAGMA) m:=80; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1 + (&+[x^(2^(k+j)): j in [0..m/4]]): k in [0..m/4]]) )); // G. C. Greubel, Feb 11 2019

CROSSREFS

Cf. A002487, A086450, A174980.

Sequence in context: A278425 A309019 A082908 * A321088 A070556 A277687

Adjacent sequences:  A086446 A086447 A086448 * A086450 A086451 A086452

KEYWORD

nonn,easy,look,tabf

AUTHOR

Ralf Stephan, Jul 20 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 05:14 EST 2019. Contains 329839 sequences. (Running on oeis4.)