A003038 - OEIS

A003038

Dimensions of split simple Lie algebras over any field of characteristic zero.
(Formerly M2712)

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)

OFFSET

1,1

REFERENCES

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).

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

EXAMPLE

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.

MAPLE

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:

MATHEMATICA

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 *)

PROG

(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

-- Reinhard Zumkeller, Dec 16 2012

CROSSREFS

Cf. A001066, A126581.

Subsequences, apart from some initial terms: A000217, A000384, A005563, A014105.