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

1, 0, -1, -4, -6, -11, -13, -20, -20, -29, -23, -34, -14, -27, 23, 8, 120, 103, 341, 322, 814, 793, 1795, 1772, 3796, 3771, 7841, 7814, 15978, 15949, 32303, 32272, 65008, 64975, 130477, 130442, 261478, 261441, 523547, 523508, 1047756, 1047715

Offset: 1

Author

Jonathan Vos Post, Jul 27 2009

Keywords

• easy
• sign
• changed

Comments

Lower bound for the essential dimension of algebraic groups with a nontrivial center.

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Patrick Brosnan, Zinovy Reichstein, Angelo Vistoli, Essential Dimension and Algebraic Stacks, arXiv:math/0701903 [math.AG], 2007.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,6,-2).

Programs

Mathematica

LinearRecurrence[{3,-1,-5,6,-2}, {1, 0, -1, -4, -6}, 50] (* G. C. Greubel, Dec 21 2016 *)
Table[2^Floor[(n-1)/2]-(n(n-1))/2,{n,50}] (* Harvey P. Dale, Aug 25 2025 *)

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

Formula

From R. J. Mathar, Sep 27 2009: (Start)

a(n) = 3*a(n-1) -a(n-2) -5*a(n-3) +6*a(n-4) -2*a(n-5).

G.f.: x*(-1-4*x^3+x^4+3*x)/((2*x^2-1)*(1-x)^3). (End)

Extensions

Edited (but not checked) by N. J. A. Sloane, Aug 01 2009