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

Author

Jonathan Sondow, Jun 17 2011

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A048648.

Programs

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 *)

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

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)