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A003038
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Dimensions of split simple Lie algebras over any field of characteristic zero.
(Formerly M2712)
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3
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3, 8, 10, 14, 15, 21, 24, 28, 35, 36, 45, 48, 52, 55, 63, 66, 78, 80, 91, 99, 105, 120, 133, 136, 143, 153, 168, 171, 190, 195, 210, 224, 231, 248, 253, 255, 276, 288, 300, 323, 325, 351, 360, 378, 399, 406, 435, 440, 465, 483, 496, 528, 561, 575, 595, 624, 630
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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REFERENCES
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Freeman J. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78 (1972), 635-652.
N. Jacobson, Lie Algebras. Wiley, NY, 1962; pp. 141-146.
I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal., 13 (1982), 988-1007.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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The Lie algebras in question and their dimensions are the following:
A_l: l(l+2), l >= 1,
B_l: l(2l+1), l >= 2,
C_l: l(2l+1), l >= 3,
D_l: l(2l-1), l >= 4,
G_2: 14, F_4: 52, E_6: 78, E_7: 133, E_8: 248.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
import Data.Set (deleteFindMin, fromList, insert)
a003038 n = a003038_list !! (n-1)
a003038_list = f (fromList (3 : [14, 52, 78, 133, 248]))
(drop 2 a005563_list) (drop 4 a000217_list) where
f s (x:xs) (y:ys) = m : f (x `insert` (y `insert` s')) xs ys where
(m, s') = deleteFindMin s
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004
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STATUS
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approved
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