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A007729 6th binary partition function. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

The sequence is the left-shifted vector of the production matrix M, with Lim_{k=1..inf} 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)

LINKS

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

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, Birkhaeuser Boston, Boston, MA, 1990.

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

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-Fran├žois Alcover, Mar 17 2014, after Alois P. Heinz *)

CROSSREFS

A column of A072170.

Cf. A002487, A007051.

Apart from initial terms, seems to coincide with A174868.

Sequence in context: A179509 A157007 A173509 * A174868 A268381 A186349

Adjacent sequences:  A007726 A007727 A007728 * A007730 A007731 A007732

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, May 06 2004

STATUS

approved

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Last modified July 22 10:34 EDT 2017. Contains 289669 sequences.