A143025 Period length 4: repeat [1, 8, 2, 8].

1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8

Offset: 0

Author

Paul Curtz, Oct 13 2008

Keywords

• nonn
• easy

Comments

Numerator of 1/n^2-1/(3n)^2 if n>0.

This can be generated from the transitions between principal quantum numbers n and 3n in the Hydrogen series: A005563(2), A061037(6), A061039(9), A061041(12), A061043(15), A061045(18), A061047(21), A061049(24),... (The mention of A005563(2) is somewhat a fluke to maintain the periodic pattern.)

Related to the continued fraction of (12*sqrt(55)-72)/19 = 0.89444115.. = 0+1/(1+1/(8+1/(2+...))). - R. J. Mathar, Jun 27 2011

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Crossrefs

Cf. A045944, A144437.

Programs

Magma

&cat [[1, 8, 2, 8]^^30]; // Wesley Ivan Hurt, Jul 10 2016

Maple

seq(op([1, 8, 2, 8]), n=0..50); # Wesley Ivan Hurt, Jul 10 2016

Mathematica

PadRight[{}, 120, {1, 8, 2, 8}] (* Harvey P. Dale, Jul 01 2015 *)

PARI

a(n)=[1,8,2,8][n%4+1] \\ Charles R Greathouse IV, Jun 02 2011

Formula

a(n+4) = a(n).

G.f.: (1+8*x+2*x^2+8*x^3)/(1-x^4).

From Wesley Ivan Hurt, Jul 10 2016: (Start)

a(n) = (19 - 13*I^(2*n) - I^(-n) - I^n)/4, where I = sqrt(-1).

a(n) = (19 - 2*cos(n*Pi/2) - 13*cos(n*Pi))/4. (End)

Extensions

Partially edited by R. J. Mathar, Dec 10 2008