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Trisection A061037(3*n-2) of the Balmer spectrum numerators extended to negative indices.
2

%I #28 Sep 11 2022 06:21:18

%S 0,-3,3,45,6,165,63,357,30,621,195,957,72,1365,399,1845,132,2397,675,

%T 3021,210,3717,1023,4485,306,5325,1443,6237,420,7221,1935,8277,552,

%U 9405,2499,10605,702,11877,3135,13221,870,14637,3843,16125,1056,17685,4623,19317

%N Trisection A061037(3*n-2) of the Balmer spectrum numerators extended to negative indices.

%C All terms are multiples of 3.

%H G. C. Greubel, <a href="/A174325/b174325.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) = A142600(n-1), n>1.

%F G.f.: -3*x*(7*x^10 +5*x^9 +39*x^8 +4*x^7 +74*x^6 +18*x^5 +58*x^4 +2*x^3 +15*x^2 +x -1) / ((x -1)^3*(x +1)^3*(x^2 +1)^3). - _Colin Barker_, Oct 15 2014

%F Sum_{n>=1} 1/a(n) = 11*log(3)/16 - 5*Pi/(48*sqrt(3)) - 1/4. - _Amiram Eldar_, Sep 11 2022

%t Table[Numerator[(n-2)*(n+2)/(4*n^2)],{n,-2,300,3}] (* _Vaclav Kotesovec_, Oct 15 2014 *)

%o (PARI) concat(0, Vec(-3*x*(7*x^10 +5*x^9 +39*x^8 +4*x^7 +74*x^6 +18*x^5 +58*x^4 +2*x^3 +15*x^2 +x -1) / ((x -1)^3*(x +1)^3*(x^2 +1)^3) + O(x^100))) \\ _Colin Barker_, Oct 15 2014

%o (Magma) I:=[0,-3,3,45,6,165,63,357,30,621,195,957]; [n le 12 select I[n] else 3*Self(n-4)-3*Self(n-8)+Self(n-12): n in [1..50]]; // _Vincenzo Librandi_, Oct 15 2014

%Y Cf. A061037, A142590, A142600.

%K sign,easy,frac,less

%O 0,2

%A _Paul Curtz_, Nov 27 2010