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A163417 a(n) = 2^(floor((n-1)/2)) - n*(n-1)/2.

%I A163417 #14 Aug 25 2025 17:10:11
%S A163417 1,0,-1,-4,-6,-11,-13,-20,-20,-29,-23,-34,-14,-27,23,8,120,103,341,
%T A163417 322,814,793,1795,1772,3796,3771,7841,7814,15978,15949,32303,32272,
%U A163417 65008,64975,130477,130442,261478,261441,523547,523508,1047756,1047715
%N A163417 a(n) = 2^(floor((n-1)/2)) - n*(n-1)/2.
%C A163417 Lower bound for the essential dimension of algebraic groups with a nontrivial center.
%C A163417 See Theorem 1.13, p.4. The essential dimension ed a of a (with respect to L) is the minimum of the transcendence degrees tr deg_k K taken over all fields of definition of a. Suppose k is a field of characteristic not equal to 2, and that sqrt(-1) is an element of k. If n is not divisible by 4 then a(n) <= ed Spin_n <= 2^(floor((n-1)/2)). If n is divisible by 4 then a(n) + 1 <= ed Spin_n <= 2^(floor((n-1)/2)) + 1.
%H A163417 G. C. Greubel, <a href="/A163417/b163417.txt">Table of n, a(n) for n = 1..1000</a>
%H A163417 Patrick Brosnan, Zinovy Reichstein, Angelo Vistoli, <a href="http://arXiv.org/abs/math/0701903">Essential Dimension and Algebraic Stacks</a>, arXiv:math/0701903 [math.AG], 2007.
%H A163417 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,6,-2).
%F A163417 From _R. J. Mathar_, Sep 27 2009: (Start)
%F A163417 a(n) = 3*a(n-1) -a(n-2) -5*a(n-3) +6*a(n-4) -2*a(n-5).
%F A163417 G.f.: x*(-1-4*x^3+x^4+3*x)/((2*x^2-1)*(1-x)^3). (End)
%t A163417 LinearRecurrence[{3,-1,-5,6,-2}, {1, 0, -1, -4, -6}, 50] (* _G. C. Greubel_, Dec 21 2016 *)
%t A163417 Table[2^Floor[(n-1)/2]-(n(n-1))/2,{n,50}] (* _Harvey P. Dale_, Aug 25 2025 *)
%o A163417 (PARI) Vec(x*(-1-4*x^3+x^4+3*x)/((2*x^2-1)*(1-x)^3) + O(x^50)) \\ _G. C. Greubel_, Dec 21 2016
%K A163417 easy,sign,changed
%O A163417 1,4
%A A163417 _Jonathan Vos Post_, Jul 27 2009
%E A163417 Edited (but not checked) by _N. J. A. Sloane_, Aug 01 2009