The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 Table of n, a(n) for n=0..40. Moussa Benoumhani and Messaoud Kolli, Finite topologies and partitions, JIS 13 (2010), Article 10.3.5, Lemma 6 6th line. Marc Le Brun, Email to N. J. A. Sloane, Jul 1991. 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. Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1). 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 Cf. A000027, A003451, A006918, A027656, A034877. Partial sums of A110660. Sequence in context: A137928 A293154 A144834 * A280186 A032246 A219901 Adjacent sequences: A006581 A006582 A006583 * A006585 A006586 A006587 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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