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A005898 Centered cube numbers: n^3 + (n+1)^3.
(Formerly M4616)
37
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 (list; graph; refs; listen; history; text; internal format)
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^2-2(n-1)..n^2). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.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. - 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. - Zak Seidov, Feb 08 2011

This is the order of an n-ball 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 r-domination in the Kronecker product of three cycles, IEEE Trans. Circuits and Systems-I: Fundamental Theory and Applications, vol. 49, no. 1, pp. 89 - 92, Jan. 2002.

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, 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), 4545-4558.

D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819-822.

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)/(1-4*x+6*x^2-4*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)/(z-1)**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^3-3*n^2+n)+2*n-1 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. - 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

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Last modified August 29 12:22 EDT 2014. Contains 246188 sequences.