A144437 - OEIS

%I #47 Dec 12 2023 08:20:58

%S 3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,

%T 3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,

%U 1,3,3,1,3,3,1,3,3,1,3,3,1,3,3,1,3,3

%N Period 3: repeat [3, 3, 1].

%C The sequence is generated from numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...

%C Conjecture: a(n) is the separatix. See A045944.

%C Also the decimal expansion of the constant 3310/999. - _R. J. Mathar_, May 21 2009

%C Continued fraction expansion of A171417.

%C Greatest common divisor of (n+1)^2-1 and (n+1)^2+2. - _Bruno Berselli_, Mar 08 2017

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).

%F a(n) = (7-4*cos(2*Pi*n/3))/3. - _Jaume Oliver Lafont_, Nov 23 2008

%F G.f.: x*(3 + 3*x + x^2)/((1 - x)*(1 + x + x^2)). - _R. J. Mathar_, May 21 2009

%F a(n) = 3/gcd(n,3). - _Reinhard Zumkeller_, Oct 30 2009

%F a(n) = denominator(n^k/3), where k>0 is an integer. - _Enrique Pérez Herrero_, Oct 05 2011

%F a(n) = gcd(T(n+1), T(2)) = A256095(n+1, 2), with the triangular numbers T = A000217, for n >= 1. - _Wolfdieter Lang_, Mar 17 2015

%F a(n) = a(n-3) for n>3; a(n) = A169609(n) for n>0. - _Wesley Ivan Hurt_, Jul 02 2016

%F E.g.f.: (1/3)*(7*exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2) - 3). - _G. C. Greubel_, Aug 24 2017

%F From _Nicolas Bělohoubek_, Nov 11 2021: (Start)

%F a(n) = 9/(a(n-2)*a(n-1)).

%F a(n) = 7 - a(n-2) - a(n-1). See also A052901 or A069705. (End)

%p seq(op([3, 3, 1]), n=1..50); # _Wesley Ivan Hurt_, Jul 02 2016

%t A144437[n_]:=Denominator[n/3]; Array[A144437,100] (* _Enrique Pérez Herrero_, Oct 05 2011 *)

%t CoefficientList[Series[(3 + 3 x + x^2)/(1 - x^3), {x, 0, 120}], x] (* _Michael De Vlieger_, Jul 02 2016 *)

%t Table[Mod[2*n^2 + 1, 3,1], {n,1,50}] (* _G. C. Greubel_, Aug 24 2017 *)

%o (PARI) a(n)=if(n%3,3,1) \\ _Charles R Greathouse IV_, Sep 28 2015

%o (Magma) &cat [[3, 3, 1]^^30]; // _Wesley Ivan Hurt_, Jul 02 2016

%Y Cf. A000217, A045944, A109007, A164306, A167192, A169609, A256095.

%Y Numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...

%K nonn,easy

%O 1,1

%A _Paul Curtz_, Oct 05 2008

%E Edited by _R. J. Mathar_, May 21 2009