0,3

3-dimensional analogue of centered polygonal numbers.

Burnside group B(3,n) has order 3^a(n).

Answer to the question: if you have a tall building and 3 plates and you need to find the highest story, a plate thrown from which does not break, what is the number of stories you can handle given n tries? - Leonid A. Broukhis (leob(AT)mailcom.com), Oct 24 2000

W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Laurent Saloff-Coste, Random walks on finite groups, in Probability on discrete structures, 263-346, Encyclopaedia Math. Sci., 110, Springer, 2004).

binomial(n+2,n-1)-binomial(n,n-2). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2006

a(n)=a(n-1)+n^2/2-n/2+1, with a(0)=0 - Paolo P. Lava (ppl(AT)spl.at), Apr 12 2007

Euler transform of length 6 sequence [ 3, 1, 1, 0, 0, -1]. - Michael Somos May 04 2007

G.f.: x*(x^2-x+1)/(1-x)^4. E.g.f.: (x+x^2/2+x^3/6) * exp(x). a(-n) = -a(n).

Starting (1, 3, 7, 14,...) = binomial transform of [1, 2, 2, 1, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 24 2008

seq(sum(binomial(n, k), k=1..3), n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 13 2007

(PARI) {a(n)= n*(n^2+5)/6} /* Michael Somos May 04 2007 */

Cf. A051576, A055795, A006552. Differences give A000217 + 1.

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Sequence in context: A123386 A060999 A089187 this_sequence A089240 A057524 A011795

Adjacent sequences: A004003 A004004 A004005 this_sequence A004007 A004008 A004009

nonn,nice,easy

Albert D. Rich (Albert_Rich(AT)msn.com).