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 A004006 a(n) = C(n,1) + C(n,2) + C(n,3), or n*(n^2 + 5)/6. 56
 0, 1, 3, 7, 14, 25, 41, 63, 92, 129, 175, 231, 298, 377, 469, 575, 696, 833, 987, 1159, 1350, 1561, 1793, 2047, 2324, 2625, 2951, 3303, 3682, 4089, 4525, 4991, 5488, 6017, 6579, 7175, 7806, 8473, 9177, 9919, 10700, 11521, 12383, 13287, 14234, 15225 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 3-dimensional analog of centered polygonal numbers. The 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 Broukhis, Oct 24 2000 Equals row sums of triangle A144329 starting with "1". - Gary W. Adamson, Sep 18 2008 Let A be the Hessenberg matrix of order n, defined by: A[1,j]=A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n-1)=-coeff(charpoly(A,x),x^(n-3)). - Milan Janjic, Jan 24 2010 From J. M. Bergot, Aug 03 2011: (Start) If one formed the 3 X 3 square   |  n  |  n+1 | n+2 |   | n+3 |  n+4 | n+5 |   | n+6 |  n+7 | n+8 |   and found the sum of the horizontal products n*(n + 1)*(n + 2) + (n + 3)*(n + 4)*(n + 5) + (n + 6)*(n + 7)*(n + 8) and added the sum of the vertical products n*(n + 3)*(n + 6) + (n + 1)*(n + 4)*(n + 7) + (n + 2)*(n + 5)(n + 8) one gets 6*n^3 + 72*n^2 + 318*n + 504. This will give 36 times the values of all the terms in this sequence. (End) a(n) is divisible by n for n congruent to {1,5} mod 6. (see A007310). - Gary Detlefs, Dec 08 2011 REFERENCES 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). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..5000 Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. [From Parthasarathy Nambi, Sep 30 2009] 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. M. Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8 Laurent Saloff-Coste, Random walks on finite groups, in Probability on discrete structures, 263-346, Encyclopaedia Math. Sci., 110, Springer, 2004). Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1). FORMULA 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). a(n) = binomial(n+2,n-1) - binomial(n,n-2). - Zerinvary Lajos, May 11 2006 a(n) = a(n-1) + n^2/2 - n/2 + 1, with a(0)=0. - Paolo P. Lava, Apr 12 2007 Euler transform of length 6 sequence [ 3, 1, 1, 0, 0, -1]. - Michael Somos, May 04 2007 Starting (1, 3, 7, 14, ...) = binomial transform of [1, 2, 2, 1, 0, 0, 0, ...]. - Gary W. Adamson, Apr 24 2008 a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Aug 21 2011 a(n) = A000292(n) + n + 1. - Reinhard Zumkeller, Mar 31 2012 a(n) = 2*a(n-1) + (n-1) - a(n-2) with a(0) = 0, a(1) = 1. - Richard R. Forberg, Jan 23 2014 a(n) = Sum_{i=1..n} binomial(n-2i,2). - Wesley Ivan Hurt, Nov 18 2017 MAPLE A004006 := proc(n) n*(n^2+5)/6 ; end proc: seq(A004006(n), n=0..10) ; # R. J. Mathar, Jun 05 2011 MATHEMATICA a=2; s=3; lst={0, 1, s}; Do[a+=n; s+=a; AppendTo[lst, s], {n, 2, 6!, 1}]; lst (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *) Table[Total[Table[Binomial[n, i], {i, 3}]], {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 3, 7}, 50] (* Harvey P. Dale, Aug 21 2011 *) PROG (PARI) {a(n)= n*(n^2+5)/6} /* Michael Somos, May 04 2007 */ (MAGMA) [n*(n^2+5)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011 (Haskell) a004006 n = a000292 n + n + 1  -- Reinhard Zumkeller, Mar 31 2012 (Maxima) A004006(n):=n*(n^2+5)/6\$ makelist(A004006(n), n, 0, 30); /* Martin Ettl, Jan 08 2013 */ CROSSREFS Cf. A051576, A055795, A006552. Differences give A000217 + 1 or A000124. 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. Cf. A144329. Cf. A228074. Sequence in context: A179178 A171973 A253895 * A089240 A057524 A293467 Adjacent sequences:  A004003 A004004 A004005 * A004007 A004008 A004009 KEYWORD nonn,nice,easy AUTHOR Albert D. Rich (Albert_Rich(AT)msn.com) STATUS approved

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