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

0,2

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A010872, A061037.

Sequence in context: A214037 A321281 A048933 * A346064 A291469 A375459

Adjacent sequences: A142587 A142588 A142589 * A142591 A142592 A142593

Keyword

nonn,easy

Author

Paul Curtz, Sep 22 2008

Extensions

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

Status

approved