Search: id:A004006
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%I A004006
%S A004006 0,1,3,7,14,25,41,63,92,129,175,231,298,377,469,575,696,833,987,1159,
%T A004006 1350,1561,1793,2047,2324,2625,2951,3303,3682,4089,4525,4991,5488,6017,
%U A004006 6579,7175,7806,8473,9177,9919,10700,11521,12383,13287,14234,15225
%N A004006 C(n,1)+C(n,2)+C(n,3), or n*(n^2+5)/6.
%C A004006 3-dimensional analogue of centered polygonal numbers.
%C A004006 Burnside group B(3,n) has order 3^a(n).
%C A004006 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
%C A004006 Equals row sums of triangle A144329 starting with "1". [From Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Sep 18 2008]
%D A004006 W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 
               1966, see p. 380.
%D A004006 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. 
               (11).
%D A004006 Michael Boardman, "The Egg-Drop Numbers", Mathematics Magazine , 77 (2004), 
               368-372. [From Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Sep 
               30 2009]
%H A004006 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A004006 N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/
               abs/math.NT/0509316">On the Integrality of n-th Roots of Generating 
               Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H A004006 Laurent Saloff-Coste, <a href="http://www.math.cornell.edu/~lsc/chap5.pdf">
               Random walks on finite groups</a>, in Probability on discrete structures, 
               263-346, Encyclopaedia Math. Sci., 110, Springer, 2004).
%F A004006 binomial(n+2,n-1)-binomial(n,n-2). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 11 2006
%F A004006 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
%F A004006 Euler transform of length 6 sequence [ 3, 1, 1, 0, 0, -1]. - Michael 
               Somos May 04 2007
%F A004006 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).
%F A004006 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
%p A004006 seq(sum(binomial(n,k),k=1..3),n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 13 2007
%t A004006 a=2;s=3;lst={0,1,s};Do[a+=n;s+=a;AppendTo[lst,s],{n,2,6!,1}];lst [From 
               Vladimir Orlovsky (4vladimir(AT)gmail.com), May 24 2009]
%o A004006 (PARI) {a(n)= n*(n^2+5)/6} /* Michael Somos May 04 2007 */
%Y A004006 Cf. A051576, A055795, A006552. Differences give A000217 + 1.
%Y A004006 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.
%Y A004006 A144329 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 18 2008]
%Y A004006 Sequence in context: A123386 A060999 A089187 this_sequence A089240 A057524 
               A011795
%Y A004006 Adjacent sequences: A004003 A004004 A004005 this_sequence A004007 A004008 
               A004009
%K A004006 nonn,nice,easy
%O A004006 0,3
%A A004006 Albert D. Rich (Albert_Rich(AT)msn.com).

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