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%I #17 Nov 21 2013 13:11:11
%S 3,6,8,10,14,15,16,20,21,24,28,30,35,36,42,45,48,52,55,56,63,66,70,72,
%T 78,80,90,91,96,99,104,105,110,120,126,132,133,136,143,153,156,160,
%U 168,171,182,190,195,198,210,224,231,240,248,253,255,266,272,276,286,288,300,306
%N Dimensions (sorted, with duplicates removed) of real simple Lie algebras.
%C The possible dimensions of real simple Lie algebras are the numbers n and 2n where n runs through the dimensions of the complex simple Lie algebras.
%D Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652.
%D N. Jacobson, Lie Algebras. Wiley, NY, 1962; see pp. 141-146.
%H Reinhard Zumkeller, <a href="/A001066/b001066.txt">Table of n, a(n) for n = 1..10000</a>
%F Numbers n and 2n as n runs through A003038.
%e 6 is the dimension of the real simple Lie algebra SL_2(C).
%t max = 18; sa = Table[k*(k+2), {k, 1, max}]; sb = Table[k*(2k+1), {k, 2, max}]; sd := Table[k*(2k-1), {k, 4, max}]; se = {14, 52, 78, 133, 248}; Select[ Union[sa, 2*sa, sb, 2*sb, sd, 2*sd, se, 2*se], # <= max^2 &] (* _Jean-François Alcover_, Apr 02 2012, after A003038 *)
%o (Haskell)
%o import Data.Set (deleteFindMin, fromList, insert)
%o a001066 n = a001066_list !! (n-1)
%o a001066_list = f (fromList [h, 2 * h]) $ tail a003038_list where
%o h = head a003038_list
%o f s (x:xs) = m : f (x `insert` (( 2 * x) `insert` s')) xs where
%o (m, s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, Dec 16 2012
%Y Cf. A003038.
%Y Subsequences, apart from some initial terms: A000217, A000384, A002378, A005563, A014105.
%K nonn,nice,easy
%O 1,1
%A Richard E. Borcherds (reb(AT)math.berkeley.edu)
%E Entry revised by _N. J. A. Sloane_, Mar 16 2007
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