A174325 Trisection A061037(3*n-2) of the Balmer spectrum numerators extended to negative indices.

0, -3, 3, 45, 6, 165, 63, 357, 30, 621, 195, 957, 72, 1365, 399, 1845, 132, 2397, 675, 3021, 210, 3717, 1023, 4485, 306, 5325, 1443, 6237, 420, 7221, 1935, 8277, 552, 9405, 2499, 10605, 702, 11877, 3135, 13221, 870, 14637, 3843, 16125, 1056, 17685, 4623, 19317

Offset: 0

Author

Paul Curtz, Nov 27 2010

Keywords

• sign
• easy
• frac
• less

Comments

All terms are multiples of 3.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Crossrefs

Cf. A061037, A142590, A142600.

Programs

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

Mathematica

Table[Numerator[(n-2)*(n+2)/(4*n^2)],{n,-2,300,3}] (* Vaclav Kotesovec, Oct 15 2014 *)

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

Formula

a(n) = A142600(n-1), n>1.

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

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