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A008676
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Expansion of 1/((1-x^3)*(1-x^5)).
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5
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1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5
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OFFSET
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0,16
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COMMENTS
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a(n) gives the number of partitions of n using only the parts 3 and 5. e.g. a(25)=2: 5+5+5+5+5 and 5+5+3+3+3+3+3+3. - Andrew Baxter, Jun 20 2011
a(n) gives the number of partitions of n+8 involving both a 3 and a 5. e.g. a(25)=2 and we may write 33 as 5+5+5+5+5+5+3 and 5+5+5+3+3+3+3+3+3. 11*3 doesn't count as no 5 is involved. - Jon Perry, Jul 03 2004
Conjecture: a(n) = Floor(2*(n + 3)/3) - Floor(3*(n + 3)/5). - John W. Layman, Sep 23 2009
Also, it appears that a(n) gives the number of distinct multisets of n-1 integers, each of which is -2, +3, or +4, such that the sum of the members of each multiset is 2. E.g., for n=5, the multiset {-2,-2,3,3}, and no others, of n-1=4 members, sums to 2, so a(5)=1. - John W. Layman, Sep 23 2009
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 217
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,0,-1).
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FORMULA
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G.f.: 1/( (1-x^3) * (1-x^5) ).
a(n) = a(n-3) + a(n-5) - a(n-8), a(0)=a(3)=a(5)=a(6)=1, a(1)=a(2)=a(4) =a(6)=a(7)=0.
a(n) = floor((2*n+5)/5) - floor((n+2)/3). - Tani Akinari, Aug 07 2013
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MAPLE
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a := proc (n) option remember; if n < 0 then return 0 elif n = 0 then return 1 else return a(n-3)+a(n-5)-a(n-8) end if end proc
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MATHEMATICA
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CoefficientList[Series[1/((1-x^3)(1-x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 23 2013 *)
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PROG
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(PARI) Vec(O(x^99)+1/(1-x^3)/(1-x^5)) \\ Charles R Greathouse IV, Jun 20 2011
(MAGMA) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( 1/((1-x^3)*(1-x^5)) )); // G. C. Greubel, Sep 08 2019
(Sage)
def A008676_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/((1-x^3)*(1-x^5))).list()
A008676_list(100) # G. C. Greubel, Sep 08 2019
(GAP) a:=[1, 0, 0, 1, 0, 1, 1, 0];; for n in [9..100] do a[n]:=a[n-3]+a[n-5]-a[n-8]; od; a; # G. C. Greubel, Sep 08 2019
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CROSSREFS
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Cf. A103221.
Sequence in context: A006928 A087890 A245077 * A025893 A296977 A025878
Adjacent sequences: A008673 A008674 A008675 * A008677 A008678 A008679
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Edited by Andrew Baxter, Jun 20, 2011
Typo in name fixed by Vincenzo Librandi, Jun 23 2013
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STATUS
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approved
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