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%I #58 Mar 26 2024 18:28:07
%S 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,
%T 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,
%U 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3
%N Periodic sequence: Repeat 2,3.
%C a(n) = smallest prime divisor of n!! for n >= 2. For biggest prime divisor of n!! see A139421. - _Artur Jasinski_, Apr 21 2008
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=-charpoly(A,-2). - _Milan Janjic_, Jan 27 2010
%C Simple continued fraction of 1+sqrt(5/3) = A176020. - _R. J. Mathar_, Mar 08 2012
%C p(n) = a(n-1) is the Abelian complexity function of the Thue-Morse word A010060. - _Nathan Fox_, Mar 12 2013
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=466">Encyclopedia of Combinatorial Structures 466</a>
%H G. Richomme, K. Saari, L. Q. Zamboni, <a href="http://dx.doi.org/10.1112/jlms/jdq063">Abelian complexity in minimal subshifts</a>, J. London Math. Soc. 83(1) (2011) 79-95.
%H G. Richomme, K. Saari, L. Q. Zamboni, <a href="http://arxiv.org/abs/0911.2914">Abelian complexity in minimal subshifts</a>, arXiv:0911.2914 [math.CO], 2009.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).
%F a(n) = 5/2 - ((-1)^n)/2.
%F a(n) = 2 + (n mod 2) = A007395(n) + A000035(n). - _Reinhard Zumkeller_, Mar 23 2005
%F a(n) = A020639(A016767(n)) for n>0. - _Reinhard Zumkeller_, Jan 29 2009
%F From _Jaume Oliver Lafont_, Mar 20 2009: (Start)
%F G.f.:(2+3*x)/(1-x^2).
%F Linear recurrence: a(0)=2, a(1)=3, a(n)=a(n-2) for n>=2. (End)
%F a(n) = A001615(2n)/A001615(n) for n > 0. - _Enrique Pérez Herrero_, Jun 06 2012
%F a(n) = floor((n+1)*5/2) - floor((n)*5/2). - _Hailey R. Olafson_, Jul 23 2014
%F a(n) = 3 - ((n+1) mod 2). - _Wesley Ivan Hurt_, Jul 24 2014
%p A010693:=n->2+(n mod 2): seq(A010693(n), n=0..100); # _Wesley Ivan Hurt_, Jul 24 2014
%t Table[5/2 - (-1)^n/2, {n, 0, 100}] or a = {}; Do[b = First[First[FactorInteger[n!! ]]]; AppendTo[a, b], {n, 2, 1000}]; a (* _Artur Jasinski_, Apr 21 2008 *)
%t 2 + Mod[Range[0, 100], 2] (* _Wesley Ivan Hurt_, Jul 24 2014 *)
%t PadRight[{},120,{2,3}] (* _Harvey P. Dale_, Jan 20 2023 *)
%o (Haskell)
%o a010693 = (+ 2) . (`mod` 2) -- _Reinhard Zumkeller_, Nov 27 2012
%o a010693_list = cycle [2,3] -- _Reinhard Zumkeller_, Mar 29 2012
%o (Magma) [2 + (n mod 2) : n in [0..100]]; // _Wesley Ivan Hurt_, Jul 24 2014
%o (PARI) a(n)=3 - (n+1)%2 \\ _Charles R Greathouse IV_, May 09 2016
%Y Cf. A139421.
%Y Cf. A026549 (partial products).
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
%E Definition rewritten by _Bruno Berselli_, Sep 30 2011
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