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 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]. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 4 08:29 EST 2021. Contains 349480 sequences. (Running on oeis4.)