|
| |
|
|
A004006
|
|
C(n,1) + C(n,2) + C(n,3), or n*(n^2+5)/6.
|
|
39
| |
|
|
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; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| 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
Equals row sums of triangle A144329 starting with "1". [From 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)). [From Milan R. Janjic (agnus(AT)blic.net), 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) [From 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).
Michael Boardman, "The Egg-Drop Numbers", Mathematics Magazine, 77 (2004), 368-372. [From Parthasarathy Nambi, Sep 30 2009]
|
|
|
LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,-6,4,-1).
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).
|
|
|
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 (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, 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]
|
|
|
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 [From Vladimir 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] (* From 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
|
|
|
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.
A144329 [From Gary W. Adamson, Sep 18 2008]
Sequence in context: A089187 A179178 A171973 * A089240 A057524 A011795
Adjacent sequences: A004003 A004004 A004005 * A004007 A004008 A004009
|
|
|
KEYWORD
| nonn,nice,easy
|
|
|
AUTHOR
| Albert D. Rich (Albert_Rich(AT)msn.com).
|
| |
|
|
