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%I #21 Sep 11 2022 06:21:33
%S 0,21,15,117,12,285,99,525,42,837,255,1221,90,1677,483,2205,156,2805,
%T 783,3477,240,4221,1155,5037,342,5925,1599,6885,462,7917,2115,9021,
%U 600,10197,2703,11445,756,12765,3363,14157,930,15621,4095,17157,1122,18765,4899,20445
%N First trisection of A061037 (Balmer line series of the hydrogen atom).
%C All terms are multiples of 3.
%H G. C. Greubel, <a href="/A142590/b142590.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,0,0,-3,0,0,0,1).
%F a(n) = A061037(2+3n).
%F a(n) mod 9 = 3*A010872(n).
%F 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
%F a(n) = 3*n*(3*n+4)*(37-27*cos(n*Pi)-6*cos(n*Pi/2))/64. - _Luce ETIENNE_, Mar 31 2017
%F Sum_{n>=1} 1/a(n) = 5/4 - 5*Pi/(48*sqrt(3)) - 11*log(3)/16. - _Amiram Eldar_, Sep 11 2022
%t Table[Numerator[1/4 - 1/#^2] &[2 + 3 n], {n, 0, 47}] (* _Michael De Vlieger_, Apr 02 2017 *)
%t 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 *)
%o (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
%o (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
%Y Cf. A010872, A061037.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Sep 22 2008
%E Edited and extended by _R. J. Mathar_, Sep 22 2008
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