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A014695 Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of Q_8.

%I A014695 #47 Feb 16 2025 08:32:33
%S A014695 1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,
%T A014695 2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,
%U A014695 1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1,2,2,1,1
%N A014695 Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of Q_8.
%C A014695 From _Klaus Brockhaus_, May 14 2010: (Start)
%C A014695 Periodic sequence: Repeat 1, 2, 2, 1.
%C A014695 a(n) = A130658(n+1).
%C A014695 Continued fraction expansion of (5+sqrt(221))/14.
%C A014695 Decimal expansion of 37/303. (End)
%H A014695 A. Adem, <a href="http://www.ams.org/notices/199707/adem.pdf">Recent developments in the cohomology of finite groups</a>, Notices Amer. Math. Soc., 44 (1997), 806-812.
%H A014695 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimpleGraph.html">Simple Graph</a>
%H A014695 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A014695 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,1).
%F A014695 G.f.: (1+x+x^2)/((1-x)*(1+x^2)) =  (1+2*x+2*x^2+x^3)/(1-x^4).
%F A014695 a(n) = (3-sqrt(2)*cos((2*n+1)*Pi/4))/2. - _Jaume Oliver Lafont_, Nov 28 2009
%F A014695 a(n) = (6-(1+i)*i^n-(1-i)*(-i)^n)/4 where i = sqrt(-1). - _Klaus Brockhaus_, May 14 2010
%F A014695 a(n) = denominator of Sum_{k=0..n} k/2. - _Arkadiusz Wesolowski_, Aug 09 2012
%t A014695 Table[Denominator[n*(n + 1)/4], {n, 0, 104}] (* _Arkadiusz Wesolowski_, Aug 09 2012 *)
%t A014695 LinearRecurrence[{1,-1,1},{1,2,2},120] (* _Harvey P. Dale_, Jan 19 2020 *)
%o A014695 (PARI) x='x+O('x^100); Vec((1+2*x+2*x^2+x^3)/(1-x^4)) \\ _Altug Alkan_, Dec 24 2015
%o A014695 (Python)
%o A014695 def A014695(n): return (1,2,2,1)[n&3] # _Chai Wah Wu_, Apr 17 2023
%Y A014695 Denominators for the sequence whose numerators are A064038.
%Y A014695 Cf. A130658, A177841. - _Klaus Brockhaus_, May 14 2010
%K A014695 easy,nonn
%O A014695 0,2
%A A014695 _N. J. A. Sloane_
%E A014695 More terms from _Klaus Brockhaus_, May 14 2010