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%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
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