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A062051 Number of partitions of n into powers of 3. 21

%I

%S 1,1,1,2,2,2,3,3,3,5,5,5,7,7,7,9,9,9,12,12,12,15,15,15,18,18,18,23,23,

%T 23,28,28,28,33,33,33,40,40,40,47,47,47,54,54,54,63,63,63,72,72,72,81,

%U 81,81,93,93,93,105,105,105,117,117,117,132,132,132,147,147,147,162

%N Number of partitions of n into powers of 3.

%C Number of different partial sums of 1+[1,*3]+[1,*3]+..., where [1,*3] means we can either add 1 or multiply by 3. E.g., a(6)=3 because we have 6=1+1+1+1+1+1=(1+1)*3=1*3+1+1+1. - _Jon Perry_, Jan 01 2004

%C Also number of partitions of n into distinct 3-smooth parts. E.g., a(10) = #{9+1, 8+2, 6+4, 6+3+1, 4+3+2+1} = #{9+1, 3+3+3+1, 3+3+1+1+1+1, 3+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 5. - _Reinhard Zumkeller_, Apr 07 2005

%C Starts to differ from A008650 at a(81). - _R. J. Mathar_, Jul 31 2010

%C If m=ceiling(log_3(2k)) and n=(3^m+1)/2-k for k in the range (3^(m-1)+1)/2+(3^(m-2))<=k<=(3^m-1)/2, this sequence gives the number of "feasible" partitions described in the sequence A254296. For instance, the terms starting at 121st term of A254296 backwards to 68th term of A254296 provide the first 54 terms of this sequence. - _Md. Towhidul Islam_, Mar 01 2015

%C From _Gary W. Adamson_, Sep 03 2016: (Start)

%C Let M =

%C 1, 0, 0, 0, 0, ...

%C 1, 0, 0, 0, 0, ...

%C 1, 0, 0, 0, 0, ...

%C 1, 1, 0, 0, 0, ...

%C 1, 1, 0, 0, 0, ...

%C 1, 1, 0, 0, 0, ...

%C 1, 1, 1, 0, 0, ...

%C 1, 1, 1, 0, 0, ...

%C ..., where the leftmost column is all 1's, and all other columns are 1's shifted down thrice. Lim_{k=1..inf} M^k has a single nonzero column, which gives the sequence. (End)

%H Seiichi Manyama, <a href="/A062051/b062051.txt">Table of n, a(n) for n = 0..1000</a>

%H Vassil Dimitrov, Laurent Imbert, and Andrew Zakaluzny, <a href="http://www.lirmm.fr/~imbert/pdfs/constmult_arith18.pdf">Multiplication by a Constant is Sublinear</a>, 18th IEEE Symposium on Computer Arithmetic (2007). See Theorem 1.

%H Md Towhidul Islam and Md Shahidul Islam, <a href="http://arxiv.org/abs/1502.07730">Number of Partitions of an n-kilogram Stone into Minimum Number of Weights to Weigh All Integral Weights from 1 to n kg(s) on a Two-pan Balance</a>, arXiv:1502.07730 [math.CO], 2015.

%H M. Latapy, <a href="https://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAA0116.1.html">Partitions of an integer into powers</a>, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.

%H M. Latapy, <a href="/A005706/a005706.pdf">Partitions of an integer into powers</a>, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]

%F a(n) = A005704([n/3]).

%F G.f.: Product_{k>=0} 1/(1-x^(3^k)). - _R. J. Mathar_, Jul 31 2010

%F If m = ceiling(log_3(2k)), define n = (3^m + 1)/2 - k for k in the range (3^(m-1)+1)/2 + (3^(m-2)) <= k <= (3^m-1)/2. Then, a(n) = Sum_{s=ceiling((k-1)/3)..(3^(m-1)-1)/2} a(s). This gives the first 2(3^(m-1))/3 terms. - _Md. Towhidul Islam_, Mar 01 2015

%F G.f.: 1 + Sum_{i>=0} x^(3^i) / Product_{j=0..i} (1 - x^(3^j)). - _Ilya Gutkovskiy_, May 07 2017

%e a(4) = 2 and the partitions are 3+1, 1+1+1+1;

%e a(9) = 5 and the partitions are 9; 3+3+3; 3+3+1+1+1; 3+1+1+1+1+1+1; 1+1+1+1+1+1+1+1+1.

%t nn=70;a=Product[1/(1-x^(3^i)),{i,0,4}];CoefficientList[Series[a,{x,0,nn}],x] (* _Geoffrey Critzer_, Oct 30 2012 *)

%o (PARI) { n=15; v=vector(n); for (i=1,n,v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2,n, k=length(v[i-1]); for (j=1,k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*3)); c=vector(n); for (i=1,n, for (j=1,2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } \\ _Jon Perry_

%Y A005704 with terms repeated 3 times.

%Y Cf. A000123, A018819, A000009, A003586, A105420, A039966, A018819, A023893, A105420, A106244, A131995, A179051, A254296.

%K nonn

%O 0,4

%A _Amarnath Murthy_, Jun 06 2001

%E More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2001

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Last modified July 28 12:43 EDT 2021. Contains 346328 sequences. (Running on oeis4.)