

A005898


Centered cube numbers: n^3 + (n+1)^3.
(Formerly M4616)


34



1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, 10745, 12691, 14859, 17261, 19909, 22815, 25991, 29449, 33201, 37259, 41635, 46341, 51389, 56791, 62559, 68705, 75241, 82179, 89531, 97309, 105525
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OFFSET

0,2


COMMENTS

Write the natural numbers in groups: 1; 2,3,4; 5,6,7,8,9; 10,11,12,13,14,15,16; ..... and add the groups, i.e. a(n)=sum(i,i=n^22(n1)..n^2).  Klaus Strassburger (strass(AT)ddfi.uniduesseldorf.de), Sep 05 2001
The numbers 1, 9, 35, 91, etc. are divisible by 1, 3, 5, 7, etc. Therefore there are no prime numbers in this list. 9 is divisible by 3 and every third number after 9 is also divisible by 3. 35 is divisible by 5 and 7 and every fifth number after 35 is also divisible by 5 and every seventh number after 35 is also divisible by 7. This pattern continues indefinitely. [From Howard Berman (howard_berman(AT)hotmail.com), Nov 07 2008]
n^3+(n+1)^3=(2n+1)*(n^2+n+1), hence all terms are composite [From Zak Seidov, Feb 08 2011]
This is the order of an nball centered at a node in the Kronecker product (or direct product) of three cycles, each of whose lengths is at least 2n+2.  Pranava K. Jha, Oct 10 2011


REFERENCES

Pranava K. Jha, Perfect rdomination in the Kronecker product of three cycles, IEEE Trans. Circuits and SystemsI: Fundamental Theory and Applications, vol. 49, no. 1, pp. 89  92, Jan. 2002.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199241, eq. (10).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 45454558.
D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819822.


LINKS

T. D. Noe, Table of n, a(n) for n=0..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
Eric Weisstein's World of Mathematics, Centered Cube Number


FORMULA

a(n) = sum_{i=0..n} A005897(i), partial sums.  Jonathan Vos Post, Feb 06 2011
G.f.: (1+5*x+5*x^2+x^3)/(14*x+6*x^24*x^3+x^4).  Simon Plouffe (see MAPLE line) and Colin Barker, Jan 02 2012
a(n) = A037270(n+1)  A037270(n).  Ivan N. Ianakiev, May 13 2012


MAPLE

A005898:=(z+1)*(z**2+4*z+1)/(z1)**4; [Simon Plouffe in his 1992 dissertation.]


MATHEMATICA

a[n_]:=n^3; Table[a[n]+a[n+1], {n, 0, 100}] [From Vladimir Joseph Stephan Orlovsky, Jan 03 2009]


PROG

sage: [i^3+(i+1)^3 for i in xrange(0, 39)]  Zerinvary Lajos, Jul 03 2008


CROSSREFS

1/12*t*(2*n^33*n^2+n)+2*n1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
Cf. A003215, A000537, A000578 [From Vladimir Joseph Stephan Orlovsky, Jan 03 2009]
Sequence in context: A071398 A212099 A212100 * A034957 A180082 A002418
Adjacent sequences: A005895 A005896 A005897 * A005899 A005900 A005901


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane.


STATUS

approved



