A005898 Centered cube numbers: n^3 + (n+1)^3.

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, 114191, 123319, 132921

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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_{j=n^2-2(n-1)..n^2} j. - 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

Positive y values of 4*x^3 - 3*x^2 = y^2. - Bruno Berselli, Apr 28 2018

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 52.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• 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).
• Michael Penn, what's the pattern, Kenneth?, YouTube video, 2021.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
• Eric Weisstein's World of Mathematics, Centered Cube Number
• D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819-822.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

Partial sums of A005897.

The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

Programs

Magma

[n^3+(n+1)^3: n in [0..40]]; // Vincenzo Librandi, Dec 16 2015

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}] (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009 *)
CoefficientList[Series[(1 + 5 x + 5 x^2 + x^3)/(1 - x)^4,{x, 0, 40}], x] (* Vincenzo Librandi, Dec 16 2015 *)

PARI

a(n)=n^3 + (n+1)^3 \\ Anders Hellström, Dec 16 2015

Python

A005898_list, m = [], [12, -6, 2, 1]
for _ in range(10**2):
    A005898_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

Sage

[i^3+(i+1)^3 for i in range(0,39)] # Zerinvary Lajos, Jul 03 2008

Formula

a(n) = Sum_{i=0..n} A005897(i), partial sums. - Jonathan Vos Post, Feb 06 2011

G.f.: (x^2+4*x+1)*(1+x)/(1-x)^3. - Simon Plouffe (see MAPLE section) and Colin Barker, Jan 02 2012; edited by N. J. A. Sloane, Feb 07 2018

a(n) = A037270(n+1) - A037270(n). - Ivan N. Ianakiev, May 13 2012

a(n) = A000217(n+1)^2 - A000217(n-1)^2. - Bob Selcoe, Mar 25 2016

a(n) = A005408(n) * A002061(n+1). - Miquel Cerda, Oct 05 2016

From Ilya Gutkovskiy, Oct 06 2016: (Start)

E.g.f.: (1 + 8*x + 9*x^2 + 2*x^3)*exp(x).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

a(n) = (A081435(n))^2 - (A081435(n) - 1)^2. - Sergey Pavlov, Mar 01 2017