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Period length 4: repeat [1, 8, 2, 8].
5

%I #34 Dec 12 2023 08:31:28

%S 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,

%T 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,

%U 1,8,2,8,1,8,2,8,1,8,2,8,1,8,2,8,1,8

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

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

%C 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.)

%C 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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).

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

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

%F From _Wesley Ivan Hurt_, Jul 10 2016: (Start)

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

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

%p seq(op([1, 8, 2, 8]), n=0..50); # _Wesley Ivan Hurt_, Jul 10 2016

%t PadRight[{}, 120, {1, 8, 2, 8}] (* _Harvey P. Dale_, Jul 01 2015 *)

%o (PARI) a(n)=[1,8,2,8][n%4+1] \\ _Charles R Greathouse IV_, Jun 02 2011

%o (Magma) &cat [[1, 8, 2, 8]^^30]; // _Wesley Ivan Hurt_, Jul 10 2016

%Y Cf. A045944, A144437.

%K nonn,easy

%O 0,2

%A _Paul Curtz_, Oct 13 2008

%E Partially edited by _R. J. Mathar_, Dec 10 2008