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Search: id:A004006
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| A004006 |
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C(n,1)+C(n,2)+C(n,3), or n*(n^2+5)/6. |
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+0
34
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| 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; listen)
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OFFSET
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0,3
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COMMENT
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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 (qntmpkt(AT)yahoo.com), Sep 18 2008]
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REFERENCES
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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 (PachaNambi(AT)yahoo.com), Sep 30 2009]
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
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).
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FORMULA
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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
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MAPLE
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seq(sum(binomial(n, k), k=1..3), n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 13 2007
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MATHEMATICA
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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]
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PROGRAM
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(PARI) {a(n)= n*(n^2+5)/6} /* Michael Somos May 04 2007 */
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CROSSREFS
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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 (qntmpkt(AT)yahoo.com), Sep 18 2008]
Sequence in context: A123386 A060999 A089187 this_sequence A089240 A057524 A011795
Adjacent sequences: A004003 A004004 A004005 this_sequence A004007 A004008 A004009
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Albert D. Rich (Albert_Rich(AT)msn.com).
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