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A006584 If n mod 2 = 0 then n*(n^2-4)/12 else n*(n^2-1)/12. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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 2nd Witt transform of A000027. - R. J. Mathar, Nov 08 2008
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
Quasipolynomial of order 2. - Charles R Greathouse IV, May 14 2013
REFERENCES
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.
LINKS
Moussa Benoumhani and Messaoud Kolli, Finite topologies and partitions, JIS 13 (2010), Article 10.3.5, Lemma 6 6th line.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
FORMULA
a(n+3) = A003451(n) + A027656(n). - Yosu Yurramendi, Aug 07 2008
G.f.: 2*x^3/((1-x)^4*(1+x)^2). a(n) = 2*A006918(n-2). - R. J. Mathar, Nov 08 2008
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Jaume Oliver Lafont, Dec 05 2008
a(n) = n*(2*n^2-5-3*(-1)^n)/24. - Luce ETIENNE, Apr 03 2015
a(n) = Sum_{i=1..n} floor(i*(n-i)/2). - Wesley Ivan Hurt, May 07 2016
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
MAPLE
A006584:=n->`if`(n mod 2 = 0, n*(n^2-4)/12, n*(n^2-1)/12): seq(A006584(n), n=0..100); # Wesley Ivan Hurt, May 08 2016
MATHEMATICA
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 *)
PROG
(PARI) A006584(n)=n*(n^2-if(n%2, 1, 4))\12 \\ M. F. Hasler, May 01 2009
(PARI) a(n)=n*if(n%2, n^2-1, n^2-4)/12 \\ Charles R Greathouse IV, Aug 11 2017
CROSSREFS
Partial sums of A110660.
Sequence in context: A137928 A293154 A144834 * A280186 A032246 A219901
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified May 20 05:07 EDT 2024. Contains 372703 sequences. (Running on oeis4.)