A023359 Number of compositions (ordered partitions) of n into powers of 2.

1, 1, 2, 3, 6, 10, 18, 31, 56, 98, 174, 306, 542, 956, 1690, 2983, 5272, 9310, 16448, 29050, 51318, 90644, 160118, 282826, 499590, 882468, 1558798, 2753448, 4863696, 8591212, 15175514, 26805983, 47350056, 83639030, 147739848, 260967362

Offset

0,3

Comments

a(n) is the number of partitions of 2n into n parts, with each partition realized into non-symmetric permutations ignoring 1's. For example a(6): the partitions of 12 into 6 are: 111117 (1), 111126 (1), 111135 (1), 111144 (1), 111225 (2), 111234 (3), 111333 (1), 112233 (3), 112224 (2), 122223 (2), 222222 (1), where the number in brackets is the number of non-symmetric permutations ignoring 1's (e.g., 111234, ignore 1's -> 234 and we can also have 243 and 324, 112233->2233 or 2323 or 2332). The sum of the bracketed numbers is a(6)=18. - Jon Perry, Jun 22 2003

Equals right border of triangle A144219. Row sums of A144219 = (1, 2, 3, 6, 10, 18, ...). - Gary W. Adamson, Sep 14 2008

Equals the INVERT transform of its aerated variant. - Gary W. Adamson, Oct 29 2010

a(n) is an eigensequence for the sequence array of the Fredholm-Rueppel sequence A036987. - Paul Barry, Nov 03 2010

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 0..200 from T. D. Noe)

G. Alkauskas, Congruence properties of the function that counts compositions into powers of 2, J. Integer Sequences 13 (2010), Article: 10.5.3, 2010.

Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016. Mentions this sequence.

N. J. A. Sloane, Transforms

Formula

G.f.: 1 / (1 - Sum_{k>=0} x^(2^k)). - Joerg Arndt, Oct 21 2012

a(n) = [n=0] + Sum_{k>=0} a(n-2^k). - Len Smiley, May 07 2001

A(x) = A(x^2)/(1 - x*A(x^2)). - Paul D. Hanna, Dec 16 2002

INVERT transform of characteristic function of powers of 2, i.e., A036987 interpreted with an offset 1 instead of 0. - Antti Karttunen, Dec 12 2003

a(n) seems to be asymptotic to A*B^n where A=0.332198..., B=1.766398... - Benoit Cloitre, Dec 17 2002. More accurately: B=1.76639811455017359722848839244009973023206928795707277527828507440838434..., A=0.58679374529351144845013208294162259198824401250194713608555348278359775... - Vaclav Kotesovec, Apr 30 2014

Satisfies A(x) = 1 + A(x) * Sum_{k>=0} x^(2^k). a(m) == 1 (mod 2) when m=2^n-1, otherwise a(m) == 0 (mod 2). - Paul D. Hanna, Aug 27 2003

a(m) == 0 (mod 4) if A000120(m+2) >= 4. In general, a(m) == 0 (mod 2^N) if A000120(m+2^(N-1)) >= 2^N. - Giedrius Alkauskas, Mar 05 2010

Example

A(x) = A(x^2) + x*A(x^2)^2 + x^2*A(x^2)^3 + x^3*A(x^2)^4 + ... = 1 + x + 2x^2 + 3x^3 + 6x^4 + 10x^5 + 18x^6 + 31x^7 + ....
From Joerg Arndt, Dec 28 2012: (Start)
There are a(6)=18 compositions of 6 into powers of 2:
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 2 1 1 ]
[ 5]  [ 1 1 2 2 ]
[ 6]  [ 1 1 4 ]
[ 7]  [ 1 2 1 1 1 ]
[ 8]  [ 1 2 1 2 ]
[ 9]  [ 1 2 2 1 ]
[10]  [ 1 4 1 ]
[11]  [ 2 1 1 1 1 ]
[12]  [ 2 1 1 2 ]
[13]  [ 2 1 2 1 ]
[14]  [ 2 2 1 1 ]
[15]  [ 2 2 2 ]
[16]  [ 2 4 ]
[17]  [ 4 1 1 ]
[18]  [ 4 2 ]
(End)

Maple

a:= proc(n) option remember;
      `if`(n=0, 1, add(a(n-2^i), i=0..ilog2(n)))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 11 2014

Mathematica

CoefficientList[Series[1/(1 - Sum[x^(2^i), {i, 0, 20}]), {x, 0, 20}], x]
a[0] = 1; a[n_] := a[n] = Sum[a[n-2^k], {k, 0, Log[2, n]}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 25 2015, after Alois P. Heinz *)

Prog

(PARI) {a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=2; A = 1 /(1 / subst(A, x, x^2) - x)); polcoeff(A, n))}; /* Michael Somos, Dec 20 2002 */
(PARI)
N=66; x='x+O('x^N);
Vec( 1/(1-sum(k=0, ceil(log(N)/log(2)), x^(2^k) ) ) )
/* Joerg Arndt, Oct 21 2012 */

Crossrefs

The column sums of the table A073265.

Cf. A073267, A073202, A073288, A144219.

Sequence in context: A077930 A060945 A349904 * A357455 A357453 A082482

Adjacent sequences: A023356 A023357 A023358 * A023360 A023361 A023362

Keyword

nonn,easy,nice

Author

David W. Wilson

Extensions

Edited by Franklin T. Adams-Watters, Aug 05 2005

Status

approved