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%I #57 May 08 2023 03:25:07
%S 1,2,4,5,8,10,13,14,18,21,26,28,33,36,40,41,46,50,57,60,68,73,80,82,
%T 89,94,102,105,112,116,121,122,128,133,142,146,157,164,174,177,188,
%U 196,209,214,226,233,242,244,253,260,272,277,290,298,309,312,322,329,340,344
%N 6th binary partition function.
%C From _Gary W. Adamson_, Aug 31 2016: (Start)
%C The sequence is the left-shifted vector of the production matrix M, with lim_{k->infinity} M^k. M =
%C 1, 0, 0, 0, 0, ...
%C 2, 0, 0, 0, 0, ...
%C 2, 1, 0, 0, 0, ...
%C 1, 2, 0, 0, 0, ...
%C 0, 2, 1, 0, 0, ...
%C 0, 1, 2, 0, 0, ...
%C 0, 0, 2, 1, 0, ...
%C 0, 0, 1, 2, 0, ...
%C ...
%C 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, ...).
%C Term a((2^n) - 1) = A007051: (1, 2, 5, 14, 41, 122, ...). (End)
%C 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
%H Alois P. Heinz, <a href="/A007729/b007729.txt">Table of n, a(n) for n = 0..10000</a>
%H Michael J. Collins and David Wilson, <a href="https://arxiv.org/abs/1812.11174">Equivalence of OEIS A007729 and A174868</a>, arXiv:1812.11174 [math.CO], 2018.
%H B. Reznick, <a href="http://dx.doi.org/10.1007/978-1-4612-3464-7_29">Some binary partition functions</a>, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.
%F 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
%F a(2k) = 2*a(k-1) + a(k); a(2k+1) = 2*a(k) + a(k-1). - _Michael J. Collins_, Dec 25 2018
%p b:= proc(n) option remember;
%p `if`(n<2, n, `if`(irem(n, 2)=0, b(n/2), b((n-1)/2) +b((n+1)/2)))
%p end:
%p a:= proc(n) option remember;
%p b(n+1) +`if`(n>0, a(n-1), 0)
%p end:
%p seq(a(n), n=0..70); # _Alois P. Heinz_, Jun 21 2012
%t 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_ *)
%o (Python)
%o from itertools import accumulate, count, islice
%o from functools import reduce
%o def A007729_gen(): # generator of terms
%o 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))
%o A007729_list = list(islice(A007729_gen(),30)) # _Chai Wah Wu_, May 07 2023
%Y A column of A072170.
%Y Cf. A002487, A007051.
%Y Apart from an initial zero, coincides with A174868.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, May 06 2004
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