A246943 a(4n) = 4*n , a(2n+1) = 8*n+4 , a(4n+2) = 2*n+1.

0, 4, 1, 12, 4, 20, 3, 28, 8, 36, 5, 44, 12, 52, 7, 60, 16, 68, 9, 76, 20, 84, 11, 92, 24, 100, 13, 108, 28, 116, 15, 124, 32, 132, 17, 140, 36, 148, 19, 156, 40, 164, 21, 172, 44, 180, 23, 188, 48, 196, 25, 204, 52, 212, 27, 220, 56, 228

Offset: 0

Author

Paul Curtz, Sep 08 2014

Keywords

• nonn
• easy

Comments

Consider the denominators of the Balmer series A061038(n) = 0, 4, 1, 36, 16, 100,... (a permutation of the squares of the nonnegative numbers i.e. A000290(n)) divided by A028310(n)=1,1,2,... . The numerators are a(n). The denominators are A138191(n).

Note that A061038(3n)=9*A061038(n), n>=1.

a(3n) is divisible by the period 3 sequence: repeat 9, 3, 3.

Examples

			Numerators of a(0)=0/1=0, a(1)=4/1=4, a(2)=1/2, a(3)=36/3=12,... .
		

Links

• J. J. Balmer, Notiz über die Spectrallinien des Wasserstoffs, Annalen der Physik, vol. 261, 5 (1885) 80-87. First published June 25 1884 (Basel).
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Crossrefs

Cf. A145979, A138191, A061038, A028310, A022998.

Programs

Maple

A246943:=n->n*(19-(-1)^n*13+2*cos(n*Pi/2))/8: seq(A246943(n), n=0..100); # Wesley Ivan Hurt, Apr 18 2017

Mathematica

LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,4,1,12,4,20,3,28},60] (* Harvey P. Dale, Jun 22 2022 *)

PARI

concat(0, Vec(x*(4*x^6+x^5+12*x^4+4*x^3+12*x^2+x+4)/((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100))) \\ Colin Barker, Sep 08 2014

Formula

Numerators of A061038(n)/A028310(n).

a(2n) = A022998(n).

G.f.: x*(4*x^6+x^5+12*x^4+4*x^3+12*x^2+x+4) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - Colin Barker, Sep 08 2014

a(n) = n*(19-13*(-1)^n+(1+(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/8. - Luce ETIENNE, May 26 2015

a(n) = n*(19-(-1)^n*13+2*cos(n*Pi/2))/8. - Giovanni Resta, May 26 2015