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A006584
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If n mod 2 = 0 then n*(n^2-4)/12 else n*(n^2-1)/12.
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10
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0, 0, 0, 2, 4, 10, 16, 28, 40, 60, 80, 110, 140, 182, 224, 280, 336, 408, 480, 570, 660, 770, 880, 1012, 1144, 1300, 1456, 1638, 1820, 2030, 2240, 2480, 2720, 2992, 3264, 3570, 3876, 4218, 4560, 4940, 5320
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OFFSET
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0,4
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COMMENTS
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Graded dimension of L''/[L',L''] for the free Lie algebra on 2 generators. Let L be a free Lie algebra with 2 generators graded by the total degree. Set L'=[L,L] and L''=[L',L']. Then a(n) is equal to the dimension of the homogeneous subspace of degree n+2 in the quotient L''/[L',L'']. - Sergei Duzhin, Mar 15 2004
Also the number of 3-element subsets of {1..n+1} whose elements sum up to an odd integer, i.e., the third column of A159916: e.g. a(3)=2 corresponds to the two subsets {1,2,4} and {2,3,4} of {1..4}. - M. F. Hasler, May 01 2009
The set of magic numbers for an idealized harmonic oscillator nucleus with a biaxially deformed prolate ellipsoid shape and an oscillator ratio of 2:1. - Jess Tauber, May 13 2013
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REFERENCES
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W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.
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LINKS
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FORMULA
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E.g.f.: x*(x*(x + 3)*exp(x) - 3*sinh(x))/12. - Ilya Gutkovskiy, May 08 2016
Sum_{n>=3} 1/a(n) = 75/8 - 12*log(2). - Amiram Eldar, Sep 17 2022
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MAPLE
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MATHEMATICA
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If[EvenQ@ #, #*(#^2 - 4)/12, #*(#^2 - 1)/12] & /@ Range[0, 40] (* or *) Table[n*(2*n^2 - 5 - 3*(-1)^n)/24, {n, 0, 40}] (* Michael De Vlieger, Apr 03 2015 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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