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%I M3850 #62 Sep 08 2022 08:44:34
%S 1,5,15,35,69,121,195,295,425,589,791,1035,1325,1665,2059,2511,3025,
%T 3605,4255,4979,5781,6665,7635,8695,9849,11101,12455,13915,15485,
%U 17169,18971,20895,22945,25125,27439,29891,32485,35225,38115
%N Centered tetrahedral numbers.
%C Binomial transform of (1,4,6,4,0,0,0,...). - _Paul Barry_, Jul 01 2003
%C If X is an n-set and Y a fixed 4-subset of X then a(n-4) is equal to the number of 4-subsets of X intersecting Y. - _Milan Janjic_, Jul 30 2007
%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="/A005894/b005894.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Rep., 273 (1996), 199-241, eq. (10).
%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 Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H B. K. Teo and N. J. A. Sloane, <a href="http://neilsloane.com/doc/magic1/magic1.html">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).
%F a(n) = (2*n + 1)*(n^2 + n + 3)/3.
%F G.f.: (1+x)*(1+x^2)/(1-x)^4.
%F a(n) = C(n, 0) + 4*C(n, 1) + 6*C(n, 2) + 4*C(n, 3). - _Paul Barry_, Jul 01 2003
%F a(n) is the sum of 4 consecutive tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)*(n+2)*(n+3)/6 = A000292(n). a(n) = A000292(n-3) + A000292(n-2) + A000292(n-1) + A000292(n). - _Alexander Adamchuk_, May 20 2006
%F a(n) = binomial(n+3,n) + binomial(n+2,n-1) + binomial(n+1,n-2) + binomial(n,n-3). (modified by _G. C. Greubel_, Nov 30 2017)
%F a(n) = a(n-1) + 2*n^2 + 2, n>=1 (first differences A005893). - _Vincenzo Librandi_, Mar 27 2011
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=5, a(2)=15, a(3)=35. - _Harvey P. Dale_, Nov 03 2011
%F E.g.f.: (3 + 12*x + 9*x^2 + 2*x^3)*exp(x)/3. - _G. C. Greubel_, Nov 30 2017
%p A005894:=(z+1)*(1+z**2)/(z-1)**4; # _Simon Plouffe_ in his 1992 dissertation
%t Table[(2n+1)(n^2+n+3)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,5,15,35},40] (* _Harvey P. Dale_, Nov 03 2011 *)
%o (PARI) a(n)=(2*n+1)*(n^2+n+3)/3 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [(2*n+1)*(n^2+n+3)/3: n in [0..30]]; // _G. C. Greubel_, Nov 30 2017
%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. A000292.
%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,nice
%O 0,2
%A _N. J. A. Sloane_
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