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A142590
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First trisection of A061037 (Balmer line series of the hydrogen atom).
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6
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0, 21, 15, 117, 12, 285, 99, 525, 42, 837, 255, 1221, 90, 1677, 483, 2205, 156, 2805, 783, 3477, 240, 4221, 1155, 5037, 342, 5925, 1599, 6885, 462, 7917, 2115, 9021, 600, 10197, 2703, 11445, 756, 12765, 3363, 14157, 930, 15621, 4095, 17157, 1122, 18765, 4899, 20445
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OFFSET
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0,2
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COMMENTS
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All terms are multiples of 3.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).
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FORMULA
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G.f.: 3*x*(-15*x^8 -18*x^5 -74*x^4 -39*x^2 -5*x-7 -4*x^3 +x^10 -2*x^7 -x^9 -58*x^6)/ ((x-1)^3*(1+x)^3*(x^2+1)^3). - R. J. Mathar, Sep 22 2008
a(n) = 3*n*(3*n+4)*(37-27*cos(n*Pi)-6*cos(n*Pi/2))/64. - Luce ETIENNE, Mar 31 2017
Sum_{n>=1} 1/a(n) = 5/4 - 5*Pi/(48*sqrt(3)) - 11*log(3)/16. - Amiram Eldar, Sep 11 2022
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MATHEMATICA
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Table[Numerator[1/4 - 1/#^2] &[2 + 3 n], {n, 0, 47}] (* Michael De Vlieger, Apr 02 2017 *)
CoefficientList[Series[3*x*(-15*x^8 -18*x^5 -74*x^4 -39*x^2 -5*x-7 -4*x^3 +x^10 -2*x^7 -x^9 -58*x^6)/ ((x-1)^3*(1+x)^3*(x^2+1)^3), {x, 0, 50}], x] (* G. C. Greubel, Sep 19 2018 *)
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PROG
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(PARI) my(x='x+O('x^50)); concat([0], Vec(3*x*(-15*x^8 -18*x^5 -74*x^4 -39*x^2 -5*x-7 -4*x^3 +x^10 -2*x^7 -x^9 -58*x^6)/((x-1)^3*(1+x)^3*(x^2+1)^3))) \\ G. C. Greubel, Sep 19 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(3*x*(-15*x^8 -18*x^5 -74*x^4 -39*x^2 -5*x-7 -4*x^3 +x^10 -2*x^7 -x^9 -58*x^6)/((x-1)^3*(1+x)^3*(x^2+1)^3))); // G. C. Greubel, Sep 19 2018
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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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