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A138191 Denominator of (n-1)*n*(n+1)/12.

%I A138191 #37 Feb 16 2025 08:33:07
%S A138191 1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,
%T A138191 1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,
%U A138191 1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1,2,1,1,1
%N A138191 Denominator of (n-1)*n*(n+1)/12.
%C A138191 Proof of 4-periodicity follows from evaluating (n+3)(n+4)(n+5)/12, subtracting (n-1)n(n+1)/12 and getting n^2+4n+5 which is an integer. - _R. J. Mathar_, Mar 07 2008
%H A138191 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KirchhoffIndex.html">Kirchhoff Index</a>.
%H A138191 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).
%F A138191 From _R. J. Mathar_, Mar 07 2008: (Start)
%F A138191 a(n) = 1 + (A000292(n-1) mod 2) = a(n-4).
%F A138191 O.g.f.: -1-5/(4(x-1))+1/(4(x+1))-1/(2(x^2+1)). (End)
%F A138191 From _Amiram Eldar_, Jan 01 2023: (Start)
%F A138191 Multiplicative with a(p^e) = 2 if p = 2 and e = 1, and 1 otherwise.
%F A138191 Dirichlet g.f.: zeta(s)*(1+1/2^s-1/4^s).
%F A138191 Sum_{k=1..n} a(k) ~ (5/4)*n. (End)
%e A138191 0, 1/2, 2, 5, 10, 35/2, 28, 42, 60, 165/2, 110, 143, 182, ...
%t A138191 Table[(n^3-n)/12,{n,120}]//Denominator (* or *) PadRight[{},120,{1,2,1,1}] (* _Harvey P. Dale_, Apr 15 2019 *)
%o A138191 (Python)
%o A138191 def A138191(n): return (1,1,2,1)[n&3] # _Chai Wah Wu_, Apr 25 2024
%Y A138191 Cf. A000292, A107453, A107453, A138190.
%K A138191 nonn,frac,mult,easy
%O A138191 1,2
%A A138191 _Eric W. Weisstein_, Mar 04 2008