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%I M2712 #23 Jan 31 2022 01:13:04
%S 3,8,10,14,15,21,24,28,35,36,45,48,52,55,63,66,78,80,91,99,105,120,
%T 133,136,143,153,168,171,190,195,210,224,231,248,253,255,276,288,300,
%U 323,325,351,360,378,399,406,435,440,465,483,496,528,561,575,595,624,630
%N Dimensions of split simple Lie algebras over any field of characteristic zero.
%D Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652.
%D N. Jacobson, Lie Algebras. Wiley, NY, 1962; pp. 141-146.
%D I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal., 13 (1982), 988-1007.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A003038/b003038.txt">Table of n, a(n) for n = 1..10000</a>
%e The Lie algebras in question and their dimensions are the following:
%e A_l: l(l+2), l >= 1,
%e B_l: l(2l+1), l >= 2,
%e C_l: l(2l+1), l >= 3,
%e D_l: l(2l-1), l >= 4,
%e G_2: 14, F_4: 52, E_6: 78, E_7: 133, E_8: 248.
%p M:=4200; M2:=M^2; sa:=[seq(l*(l+2),l=1..M)]; sb:=[seq(l*(2*l+1),l=2..M)]; sd:=[seq(l*(2*l-1),l=4..M)]; se:=[14,52,78,133,248]; s:=convert(sa,set) union convert(sb,set) union convert(sd,set) union convert(se,set); t:=convert(s,list); for i from 1 to nops(t) do if t[i] <= M2 then lprint(i,t[i]); fi; od:
%t max = 26; 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, sb, sd, se], # <= max^2 &](* _Jean-François Alcover_, Nov 18 2011, after Maple *)
%o (Haskell)
%o import Data.Set (deleteFindMin, fromList, insert)
%o a003038 n = a003038_list !! (n-1)
%o a003038_list = f (fromList (3 : [14, 52, 78, 133, 248]))
%o (drop 2 a005563_list) (drop 4 a000217_list) where
%o f s (x:xs) (y:ys) = m : f (x `insert` (y `insert` s')) xs ys where
%o (m, s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, Dec 16 2012
%Y Cf. A001066, A126581.
%Y Subsequences, apart from some initial terms: A000217, A000384, A005563, A014105.
%K nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
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