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A245332
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Number of compositions of n into parts 2 and 3 with at least one 2 and one 3.
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5
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0, 0, 0, 0, 0, 2, 0, 3, 3, 4, 6, 9, 10, 16, 20, 27, 36, 49, 63, 86, 113, 150, 199, 265, 349, 465, 615, 815, 1080, 1432, 1895, 2513, 3328, 4409, 5841, 7739, 10250, 13581, 17990, 23832, 31571, 41824, 55403, 73396, 97228, 128800, 170624, 226030, 299424, 396655
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OFFSET
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0,6
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COMMENTS
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The g.f. for compositions of n into parts r, s with both parts present is 1 + 1/(1-x^r-x^s) - 1/(1-x^r) - 1/(1-x^s). - Joerg Arndt, Jul 24 2014
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LINKS
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FORMULA
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a(n) = a(n-2)+a(n-3)+b(n) with initial terms a(5)=2,a(6)=0,a(7)=3 and b(8)=1,b(9)=1,b(10)=0,b(11)=2,b(12)=0,b(13)=1 and b(n)=b(n-6).
G.f.: x^5*(x^2+2*x+2) / ((x-1)*(x+1)*(x^2+x+1)*(x^3+x^2-1)). - Colin Barker, Jul 18 2014
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EXAMPLE
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a(12) = number of rearrangements of 33222 = 5!/(3!*2!) = 10.
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MATHEMATICA
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CoefficientList[Series[x^5 (x^2 + 2 x + 2)/((x - 1) (x + 1) (x^2 + x + 1) (x^3 + x^2 - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 25 2014 *)
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PROG
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(PARI) a=vector(100); a[5]=2; a[7]=3; b=[1, 1, 0, 2, 0, 1]; k=1; for(n=8, #a, a[n]=a[n-2]+a[n-3]+b[k]; if(k==6, k=1, k++)); a \\ Colin Barker, Jul 18 2014
(PARI)
x='x+O('x^66);
r=2; s=3;
gf = 1 + 1/(1-x^r-x^s) - 1/(1-x^r) - 1/(1-x^s);
Vec(gf)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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