A142590 First trisection of A061037 (Balmer line series of the hydrogen atom).

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

Offset: 0

Author

Paul Curtz, Sep 22 2008

Keywords

• nonn
• easy

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. A010872, A061037.

Programs

Magma

m:=25; R:=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

Mathematica

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 *)

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

Formula

a(n) = A061037(2+3n).

a(n) mod 9 = 3*A010872(n).

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

Extensions

Edited and extended by R. J. Mathar, Sep 22 2008