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 A062051 Number of partitions of n into powers of 3. 17
 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, 23, 28, 28, 28, 33, 33, 33, 40, 40, 40, 47, 47, 47, 54, 54, 54, 63, 63, 63, 72, 72, 72, 81, 81, 81, 93, 93, 93, 105, 105, 105, 117, 117, 117, 132, 132, 132, 147, 147, 147, 162 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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 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 Starts to differ from A008650 at a(81). - R. J. Mathar, Jul 31 2010 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 From Gary W. Adamson, Sep 03 2016: (Start) Let M =   1, 0, 0, 0, 0,...   1, 0, 0, 0, 0,...   1, 0, 0, 0, 0,...   1, 1, 0, 0, 0,...   1, 1, 0, 0, 0,...   1, 1, 0, 0, 0,...   1, 1, 1, 0, 0,...   1, 1, 1, 0, 0,... ..., 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) LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 Vassil Dimitrov, Laurent Imbert, and Andrew Zakaluzny, Multiplication by a Constant is Sublinear, 18th IEEE Symposium on Computer Arithmetic (2007). See Theorem 1. Md Towhidul Islam & Md Shahidul Islam, 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, arXiv:1502.07730 [math.CO], 2015. M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission] FORMULA a(n) = A005704([n/3]). G.f.: Product_{k>=0} 1/(1-x^(3^k)). - R. J. Mathar, Jul 31 2010 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 G.f.: 1 + Sum_{i>=0} x^(3^i) / Product_{j=0..i} (1 - x^(3^j)). - Ilya Gutkovskiy, May 07 2017 EXAMPLE a(4) = 2 and the partitions are 3+1, 1+1+1+1; 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. MATHEMATICA nn=70; a=Product[1/(1-x^(3^i)), {i, 0, 4}]; CoefficientList[Series[a, {x, 0, nn}], x] (* Geoffrey Critzer, Oct 30 2012 *) PROG (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 CROSSREFS A005704 with terms repeated 3 times. Cf. A000123, A018819, A000009, A003586, A105420, A039966, A018819, A023893, A105420, A106244, A131995, A179051, A254296. Sequence in context: A076973 A008649 A008650 * A179269 A108711 A261736 Adjacent sequences:  A062048 A062049 A062050 * A062052 A062053 A062054 KEYWORD nonn AUTHOR Amarnath Murthy, Jun 06 2001 EXTENSIONS More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2001 STATUS approved

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Last modified October 23 14:06 EDT 2018. Contains 316528 sequences. (Running on oeis4.)