A007728 - OEIS

%I #34 Aug 02 2023 14:36:46

%S 1,1,2,2,4,3,5,4,8,6,9,7,12,8,12,9,17,12,18,14,23,15,22,16,28,19,27,

%T 20,32,20,29,21,38,26,38,29,47,30,44,32,55,37,52,38,60,37,53,38,66,44,

%U 63,47,74,46,66,47,79,52,72,52,81,49,70,50,88,59,85,64

%N 5th binary partition function.

%C The number of ways of writing n as a sum of powers of 2, each power being used at most four times. - _Dmitry Kamenetsky_, Jul 14 2023

%H Alois P. Heinz, <a href="/A007728/b007728.txt">Table of n, a(n) for n = 0..10000</a>

%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.: Product_{k>=0} (1 - x^(5*2^k))/(1 - x^(2^k)). - _Ilya Gutkovskiy_, Jul 09 2019

%p b:= proc(n, i) option remember;

%p       `if`(n=0, 1, `if`(i<0, 0, add(`if`(n-j*2^i<0, 0,

%p          b(n-j*2^i, i-1)), j=0..4)))

%p     end:

%p a:= n-> b(n, ilog2(n)):

%p seq(a(n), n=0..70);  # _Alois P. Heinz_, Jun 21 2012

%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 0, 0, Sum[If[n-j*2^i < 0, 0, b[n-j*2^i, i-1, k]], {j, 0, k-1}]]]; a[n_] := b[n, Log[2, n] // Floor, 5]; Table[a[n], {n, 0, 70} ] (* _Jean-François Alcover_, Jan 17 2014, after _Alois P. Heinz_ *)

%Y A column of A072170.

%Y Cf. A000123, A018819.

%K nonn,look

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, May 06 2004