A005898 Centered cube numbers: n^3 + (n+1)^3. (Formerly M4616)

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

0,2

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_{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).

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

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 *)

Prog

(Sage) [i^3+(i+1)^3 for i in range(0, 39)] # Zerinvary Lajos, Jul 03 2008
(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
(Magma) [n^3+(n+1)^3: n in [0..40]]; // Vincenzo Librandi, Dec 16 2015
(PARI) a(n)=n^3 + (n+1)^3 \\ Anders Hellström, Dec 16 2015

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.

Sequence in context: A071398 A212099 A212100 * A034957 A180082 A002418

Adjacent sequences: A005895 A005896 A005897 * A005899 A005900 A005901

Keyword

nonn,easy,changed

Author

N. J. A. Sloane

Status

approved