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
KEYWORD
nonn,easy
AUTHOR
Jonathan Sondow, Jun 17 2011
STATUS
approved