A007729 6th binary partition function.

1, 2, 4, 5, 8, 10, 13, 14, 18, 21, 26, 28, 33, 36, 40, 41, 46, 50, 57, 60, 68, 73, 80, 82, 89, 94, 102, 105, 112, 116, 121, 122, 128, 133, 142, 146, 157, 164, 174, 177, 188, 196, 209, 214, 226, 233, 242, 244, 253, 260, 272, 277, 290, 298, 309, 312, 322, 329, 340, 344

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

From Gary W. Adamson, Aug 31 2016: (Start)

The sequence is the left-shifted vector of the production matrix M, with lim_{k->infinity} M^k. M =

1, 0, 0, 0, 0, ...

2, 0, 0, 0, 0, ...

2, 1, 0, 0, 0, ...

1, 2, 0, 0, 0, ...

0, 2, 1, 0, 0, ...

0, 1, 2, 0, 0, ...

0, 0, 2, 1, 0, ...

0, 0, 1, 2, 0, ...

...

The sequence is equal to the product of its aerated variant by (1,2,2,1): (1, 2, 2, 1) * (1, 0, 2, 0, 4, 0, 5, 0, 8, ...) = (1, 2, 4, 5, 8, 10, ...).

Term a((2^n) - 1) = A007051: (1, 2, 5, 14, 41, 122, ...). (End)

a(n) is the number of ways to represent 2n (or 2n+1) as a sum e_0 + 2*e_1 + ... + (2^k)*e_k with each e_i in {0,1,2,3,4,5}. - Michael J. Collins, Dec 25 2018

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Michael J. Collins and David Wilson, Equivalence of OEIS A007729 and A174868, arXiv:1812.11174 [math.CO], 2018.
• B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.

Crossrefs

A column of A072170.

Cf. A002487, A007051.

Apart from an initial zero, coincides with A174868.

Programs

Maple

b:= proc(n) option remember;
      `if`(n<2, n, `if`(irem(n, 2)=0, b(n/2), b((n-1)/2) +b((n+1)/2)))
    end:
a:= proc(n) option remember;
      b(n+1) +`if`(n>0, a(n-1), 0)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jun 21 2012

Mathematica

b[n_] := b[n] = If[n<2, n, If[Mod[n, 2] == 0, b[n/2], b[(n-1)/2]+b[(n+1)/2]]]; a[n_] := a[n] = b[n+1] + If[n>0, a[n-1], 0]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

Python

from itertools import accumulate, count, islice
from functools import reduce
def A007729_gen(): # generator of terms
    return accumulate(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) for n in count(1))
A007729_list = list(islice(A007729_gen(),30)) # Chai Wah Wu, May 07 2023

Formula

G.f.: (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = (1 + 2x + 2x^2 + x^3 + 0 + 0 + 0 + ...). - Gary W. Adamson, Sep 01 2016

a(2k) = 2*a(k-1) + a(k); a(2k+1) = 2*a(k) + a(k-1). - Michael J. Collins, Dec 25 2018

Extensions

More terms from Vladeta Jovovic, May 06 2004