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%I M4616 #117 Mar 20 2023 18:52:20
%S 1,9,35,91,189,341,559,855,1241,1729,2331,3059,3925,4941,6119,7471,
%T 9009,10745,12691,14859,17261,19909,22815,25991,29449,33201,37259,
%U 41635,46341,51389,56791,62559,68705,75241,82179,89531,97309,105525,114191,123319,132921
%N Centered cube numbers: n^3 + (n+1)^3.
%C 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
%C 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
%C n^3 + (n+1)^3 = (2n+1)*(n^2+n+1), hence all terms are composite. - _Zak Seidov_, Feb 08 2011
%C 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
%C Positive y values of 4*x^3 - 3*x^2 = y^2. - _Bruno Berselli_, Apr 28 2018
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A005898/b005898.txt">Table of n, a(n) for n = 0..1000</a>
%H Pranava K. Jha, <a href="http://dx.doi.org/10.1109/81.974881">Perfect r-domination in the Kronecker product of three cycles</a>, IEEE Trans. Circuits and Systems-I: Fundamental Theory and Applications, vol. 49, no. 1, pp. 89 - 92, Jan. 2002.
%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, 273 (1996), 199-241, eq. (10).
%H Michael Penn, <a href="https://www.youtube.com/watch?v=d4Y6harH5cY">what's the pattern, Kenneth?</a>, YouTube video, 2021.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredCubeNumber.html">Centered Cube Number</a>
%H D. Zeitlin, <a href="http://www.jstor.org/stable/2319798">A family of Galileo sequences</a>, Amer. Math. Monthly 82 (1975), 819-822.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = Sum_{i=0..n} A005897(i), partial sums. - _Jonathan Vos Post_, Feb 06 2011
%F 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
%F a(n) = A037270(n+1) - A037270(n). - _Ivan N. Ianakiev_, May 13 2012
%F a(n) = A000217(n+1)^2 - A000217(n-1)^2. - _Bob Selcoe_, Mar 25 2016
%F a(n) = A005408(n) * A002061(n+1). - _Miquel Cerda_, Oct 05 2016
%F From _Ilya Gutkovskiy_, Oct 06 2016: (Start)
%F E.g.f.: (1 + 8*x + 9*x^2 + 2*x^3)*exp(x).
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
%F a(n) = (A081435(n))^2 - (A081435(n) - 1)^2. - _Sergey Pavlov_, Mar 01 2017
%p A005898:=(z+1)*(z**2+4*z+1)/(z-1)**4; # _Simon Plouffe_ in his 1992 dissertation
%t a[n_]:=n^3; Table[a[n]+a[n+1], {n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 03 2009 *)
%t CoefficientList[Series[(1 + 5 x + 5 x^2 + x^3)/(1 - x)^4,{x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 16 2015 *)
%o (Sage) [i^3+(i+1)^3 for i in range(0,39)] # _Zerinvary Lajos_, Jul 03 2008
%o (Python)
%o A005898_list, m = [], [12, -6, 2, 1]
%o for _ in range(10**2):
%o A005898_list.append(m[-1])
%o for i in range(3):
%o m[i+1] += m[i] # _Chai Wah Wu_, Dec 15 2015
%o (Magma) [n^3+(n+1)^3: n in [0..40]]; // _Vincenzo Librandi_, Dec 16 2015
%o (PARI) a(n)=n^3 + (n+1)^3 \\ _Anders Hellström_, Dec 16 2015
%Y (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.
%Y Cf. A003215, A000537, A000578. - _Vladimir Joseph Stephan Orlovsky_, Jan 03 2009
%Y Partial sums of A005897.
%Y 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.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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