%I #23 Nov 03 2019 01:46:27

%S 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,

%T 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,

%U 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

%N 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].

%C 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.

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

%H Colin Barker, <a href="/A189996/b189996.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BottPeriodicityTheorem.html">Bott Periodicity Theorem</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bott_periodicity">Bott periodicity</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Orthogonal_group#Homotopy_groups">Orthogonal group</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,1).

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

%F From _Colin Barker_, Nov 02 2019: (Start)

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

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

%F (End)

%t LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{2, 2, 1, 0, 1, 1, 1, 0},104] (* _Ray Chandler_, Aug 25 2015 *)

%t PadRight[{},120,{2,2,1,0,1,1,1,0}] (* _Harvey P. Dale_, Jun 13 2017 *)

%o (PARI) a(n)=[2, 2, 1, 0, 1, 1, 1, 0][n%8+1] \\ _Charles R Greathouse IV_, Jul 13 2016

%o (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

%Y Cf. A048648.

%K nonn,easy

%O 0,1

%A _Jonathan Sondow_, Jun 17 2011