A189996 Bott periodicity: the homotopy groups of the stable orthogonal group are periodic with period 8 and repeat like [2, 2, 1, 0, 1, 1, 1, 0].

2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0, 2, 2, 1, 0, 1, 1, 1, 0

Offset

0,1

Comments

Bott proved that the n-th homotopy group of the stable orthogonal group is Z/(a(n)*Z), where Z is the integers and Z/(0*Z), Z/(1*Z), Z/(2*Z) are the cyclic groups of order infinity, 1, 2, respectively. For details, see the Wikipedia orthogonal group link.

For references and additional links, see the Wikipedia Bott periodicity link.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Bott Periodicity Theorem

Wikipedia, Bott periodicity

Wikipedia, Orthogonal group

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Formula

a(n) = 2, 2, 1, 0, 1, 1, 1, 0 if n == 0, 1, 2, 3, 4, 5, 6, 7 (mod 8), respectively.

From Colin Barker, Nov 02 2019: (Start)

G.f.: (2 + 2*x + x^2 + x^4 + x^5 + x^6) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)).

a(n) = a(n-8) for n>7.

(End)

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1}, {2, 2, 1, 0, 1, 1, 1, 0}, 104] (* Ray Chandler, Aug 25 2015 *)
PadRight[{}, 120, {2, 2, 1, 0, 1, 1, 1, 0}] (* Harvey P. Dale, Jun 13 2017 *)

Prog

(PARI) a(n)=[2, 2, 1, 0, 1, 1, 1, 0][n%8+1] \\ Charles R Greathouse IV, Jul 13 2016
(PARI) Vec((2 + 2*x + x^2 + x^4 + x^5 + x^6) / ((1 - x)*(1 + x)*(1 + x^2)*(1 + x^4)) + O(x^90)) \\ Colin Barker, Nov 02 2019

Crossrefs

Cf. A048648.

Sequence in context: A016372 A016342 A016385 * A016390 A327688 A055800

Adjacent sequences: A189993 A189994 A189995 * A189997 A189998 A189999

Keyword

nonn,easy

Author

Jonathan Sondow, Jun 17 2011

Status

approved