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A004006 C(n,1) + C(n,2) + C(n,3), or n*(n^2+5)/6. 54
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 3x3 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.

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). [From 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

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.

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 A011795

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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Last modified May 23 16:52 EDT 2017. Contains 286925 sequences.