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A008743
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Molien series for 3-dimensional group [3,4]+ = 432.
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1
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1, 0, 1, 0, 2, 0, 3, 0, 4, 1, 5, 1, 7, 2, 8, 3, 10, 4, 12, 5, 14, 7, 16, 8, 19, 10, 21, 12, 24, 14, 27, 16, 30, 19, 33, 21, 37, 24, 40, 27, 44, 30, 48, 33, 52, 37, 56, 40, 61, 44, 65, 48, 70, 52, 75, 56, 80, 61, 85, 65, 91, 70, 96, 75, 102, 80, 108, 85, 114
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OFFSET
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0,5
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COMMENTS
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The binary quintic has four invariants of degrees 4, 8, 12, 18. Those of degrees 4, 8, 12 are algebraically independent, the one of degree 18 squares to an expression in the others. [A. E. Brouwer]
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LINKS
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FORMULA
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Euler transform of length 18 sequence [ 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1].
G.f.: (1 + x^9) / ((1 - x^2) * (1 - x^4) * (1 - x^6)).
a(-3 - n) = a(n).
G.f.: (1-x^3+x^6)/((1+x+x^2)*(1+x^2)*(1+x)^2*(1-x)^3). - R. J. Mathar, Dec 18 2014
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EXAMPLE
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G.f. = 1 + x^2 + 2*x^4 + 3*x^6 + 4*x^8 + x^9 + 5*x^10 + x^11 + 7*x^12 + 2*x^13 + 8*x^14 + ...
G.f. = 1 + q^4 + 2*q^8 + 3*q^12 + 4*q^16 + q^18 + 5*q^20 + q^22 + 7*q^24 + 2*q^26 + 8*q^28 + ...
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MAPLE
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seq(coeff(series((x^9+1)/((-x^2+1)*(-x^4+1)*(-x^6+1)), x, n+1), x, n), n = 0 .. 70); # modified by G. C. Greubel, Aug 03 2019
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MATHEMATICA
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CoefficientList[Series[(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)), {x, 0, 70}], x] (* T. D. Noe, Oct 30 2011 *)
LinearRecurrence[{0, 1, 1, 1, -1, -1, -1, 0, 1}, {1, 0, 1, 0, 2, 0, 3, 0, 4}, 70] (* Harvey P. Dale, Apr 09 2019 *)
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PROG
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(PARI) {a(n) = round( (if( n%2, n-9, n) \ 2 + 3)^2 / 12)} /* Michael Somos, Oct 30 2011 */
(PARI) {a(n) = if( n<-1, n = -3 - n); polcoeff( (1+x^9)/(1-x^2)/(1-x^4)/(1-x^6) + x * O(x^n), n)} /* Michael Somos, Oct 30 2011 */
(Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Aug 03 2019
(Sage) ((1+x^9)/((1-x^2)*(1-x^4)*(1-x^6))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019
(GAP) a:=[1, 0, 1, 0, 2, 0, 3, 0, 4];; for n in [10..70] do a[n]:=a[n-2]+a[n-3] + a[n-4]-a[n-5]-a[n-6]-a[n-7]+a[n-9]; od; a; # G. C. Greubel, Aug 03 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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