This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A006584 #59 Sep 17 2022 03:44:06
%S A006584 0,0,0,2,4,10,16,28,40,60,80,110,140,182,224,280,336,408,480,570,660,
%T A006584 770,880,1012,1144,1300,1456,1638,1820,2030,2240,2480,2720,2992,3264,
%U A006584 3570,3876,4218,4560,4940,5320
%N A006584 If n mod 2 = 0 then n*(n^2-4)/12 else n*(n^2-1)/12.
%C A006584 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
%C A006584 Also the 2nd Witt transform of A000027. - _R. J. Mathar_, Nov 08 2008
%C A006584 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
%C A006584 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
%C A006584 Quasipolynomial of order 2. - _Charles R Greathouse IV_, May 14 2013
%D A006584 W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.
%H A006584 Moussa Benoumhani and Messaoud Kolli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Benoumhani/benoumhani6.html">Finite topologies and partitions</a>, JIS 13 (2010), Article 10.3.5, Lemma 6 6th line.
%H A006584 Marc Le Brun, <a href="/A006577/a006577.pdf">Email to N. J. A. Sloane, Jul 1991</a>.
%H A006584 Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
%H A006584 Pieter Moree, <a href="http://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.
%H A006584 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A006584 a(n+3) = A003451(n) + A027656(n). - _Yosu Yurramendi_, Aug 07 2008
%F A006584 G.f.: 2*x^3/((1-x)^4*(1+x)^2). a(n) = 2*A006918(n-2). - _R. J. Mathar_, Nov 08 2008
%F A006584 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
%F A006584 a(n) = n*(2*n^2-5-3*(-1)^n)/24. - _Luce ETIENNE_, Apr 03 2015
%F A006584 a(n) = Sum_{i=1..n} floor(i*(n-i)/2). - _Wesley Ivan Hurt_, May 07 2016
%F A006584 E.g.f.: x*(x*(x + 3)*exp(x) - 3*sinh(x))/12. - _Ilya Gutkovskiy_, May 08 2016
%F A006584 Sum_{n>=3} 1/a(n) = 75/8 - 12*log(2). - _Amiram Eldar_, Sep 17 2022
%p A006584 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
%t A006584 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 *)
%o A006584 (PARI) A006584(n)=n*(n^2-if(n%2,1,4))\12 \\ _M. F. Hasler_, May 01 2009
%o A006584 (PARI) a(n)=n*if(n%2, n^2-1, n^2-4)/12 \\ _Charles R Greathouse IV_, Aug 11 2017
%Y A006584 Cf. A000027, A003451, A006918, A027656, A034877.
%Y A006584 Partial sums of A110660.
%K A006584 nonn,easy
%O A006584 0,4
%A A006584 _N. J. A. Sloane_