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A008677
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Expansion of 1/((1-x^3)*(1-x^5)*(1-x^7)).
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2
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1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 14, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 31, 33, 33, 34, 35, 35, 37, 37, 38, 39, 40, 41, 41, 43
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OFFSET
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0,11
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COMMENTS
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Number of partitions of n into parts 3, 5, and 7. - Joerg Arndt, Aug 17 2013
Number of different total numbers of kicks, tries and converted tries which lead to a score of n in a rugby (union) match. - Matthew Scroggs, Jul 09 2015
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 114, [6t].
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,1,-1,0,-1,0,-1,0,0,1).
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FORMULA
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a(n) = a(n-3) + a(n-5) + a(n-7) - a(n-8) - a(n-10) - a(n-12) + a(n-15) for n >= 15. - David Neil McGrath, Sep 03 2014
G.f.: 1 / ((1 - x^3) * (1 - x^5) * (1 - x^7)).
Euler transform of length 7 sequence [ 0, 0, 1, 0, 1, 0, 1]. - Michael Somos, Sep 30 2014
0 = a(n) - a(n+3) - a(n+5) + a(n+8) - [mod(n, 7) == 6] for all n in Z. - Michael Somos, Sep 30 2014
a(n) = round(n^2/210 + n/14 + 5/21) + r(n) where r(n) = 1 if n == 0, 3, 10, 15, 45, 75, 80, 87, or 90 (mod 105), r(n) = -1 if n == 4, 11, 16, 44, 46, 74, 79 or 86 (mod 105), r(n) = 0 otherwise. - Robert Israel, Jul 09 2015
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EXAMPLE
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G.f. = 1 + x^3 + x^5 + x^6 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + 2*x^12 + ...
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MAPLE
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S:= series(1/((1-x^3)*(1-x^5)*(1-x^7)), x, 101):
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MATHEMATICA
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CoefficientList[Series[1/((1-x^3)(1-x^5)(1-x^7)), {x, 0, 100}], x] (* Vincenzo Librandi, Jun 23 2013 *)
LinearRecurrence[{0, 0, 1, 0, 1, 0, 1, -1, 0, -1, 0, -1, 0, 0, 1}, {1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2}, 70] (* Harvey P. Dale, Aug 04 2020 *)
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PROG
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(PARI) a(n)=[1, 0, -2, 2, -2, 0, 1][n%7+1]/7 +[2, -1, 0, 0, -1][n%5+1]/5 +[2, -1, -1][n%3+1]/9 +(3*n^2+45*n+148)/630; \\ Tani Akinari, Aug 17 2013
(PARI) a(n)=floor((n^2+15*n+86)/210+(n%3<1)/3+3*(n%5<1)/5) \\ Tani Akinari, Sep 30 2014
(Magma) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( 1/((1-x^3)*(1-x^5)*(1-x^7)) )); // G. C. Greubel, Sep 08 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/((1-x^3)*(1-x^5)*(1-x^7))).list()
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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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