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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->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
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.
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
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 *)
PROG
(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
CROSSREFS
A column of A072170.
Apart from an initial zero, coincides with A174868.
Sequence in context: A179509 A157007 A173509 * A174868 A268381 A186349
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, May 06 2004
STATUS
approved

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Last modified July 23 23:47 EDT 2024. Contains 374575 sequences. (Running on oeis4.)