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%I M4384 N1844 #181 Mar 23 2024 08:14:50
%S 1,7,25,63,129,231,377,575,833,1159,1561,2047,2625,3303,4089,4991,
%T 6017,7175,8473,9919,11521,13287,15225,17343,19649,22151,24857,27775,
%U 30913,34279,37881,41727,45825,50183,54809,59711,64897,70375,76153,82239
%N Centered octahedral numbers (crystal ball sequence for cubic lattice).
%C Number of points in simple cubic lattice at most n steps from origin.
%C If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 6-subsets of X intersecting each Y_i (i=1,2,3). - _Milan Janjic_, Aug 26 2007
%C Equals binomial transform of [1, 6, 12, 8, 0, 0, 0, ...] where (1, 6, 12, 8) = row 3 of the Chebyshev triangle A013609. - _Gary W. Adamson_, Jul 19 2008
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 4, a(n-2) = -coeff(charpoly(A,x),x^(n-3)). - _Milan Janjic_, Jan 26 2010
%C a(n) = A005408(n) * A097080(n-1) / 3. - _Reinhard Zumkeller_, Dec 15 2013
%C a(n) = D(3,n) where D are the Delannoy numbers (A008288). As such, a(n) gives the number of grid paths from (0,0) to (3,n) using steps that move one unit north, east, or northeast. - _David Eppstein_, Sep 07 2014
%C The first comment above can be re-expressed and generalized as follows: a(n) is the number of points in Z^3 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array (A008288), as a special case of _Dmitry Zaitsev_'s Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 3 from any given point. - _Shel Kaphan_, Jan 02 2023
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%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="/A001845/b001845.txt">Table of n, a(n) for n = 0..1000</a>
%H Luciano Ancora, <a href="https://upload.wikimedia.org/wikipedia/commons/9/9c/FigurateN.pdf">The Square Pyramidal Number and other figurate numbers</a>, ch. 4.
%H Bela Bajnok, <a href="https://arxiv.org/abs/1705.07444">Additive Combinatorics: A Menu of Research Problems</a>, arXiv:1705.07444 [math.NT], May 2017. See Section 2.3.
%H D. Bump, K. Choi, P. Kurlberg, and J. Vaaler, <a href="http://www.cecm.sfu.ca/~choi/paper/lrh.pdf">A local Riemann hypothesis, I</a> pages 16 and 17
%H J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1098/rspa.1997.0126">Low-Dimensional Lattices VII: Coordination Sequences</a>, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences </a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
%H G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
%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 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 R. G. Stanton and D. D. Cowan, <a href="http://dx.doi.org/10.1137/1012049">Note on a "square" functional equation</a>, SIAM Rev., 12 (1970), 277-279.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HauyConstruction.html">Haüy Construction</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OctahedralNumber.html">Octahedral Number</a>
%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: (1+x)^3 /(1-x)^4. [conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation]
%F a(n) = (2*n+1)*(2*n^2 + 2*n + 3)/3.
%F First differences of A014820(n). - _Alexander Adamchuk_, May 23 2006
%F a(n) = a(n-1) + 4*n^2 + 2, a(0)=1. - _Vincenzo Librandi_, Mar 27 2011
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=1, a(1)=7, a(2)=25, a(3)=63. - _Harvey P. Dale_, Jun 05 2013
%F a(n) = Sum_{k=0..min(3,n)} 2^k * binomial(3,k) * binomial(n,k). See Bump et al. - _Tom Copeland_, Sep 05 2014
%F From _Luciano Ancora_, Jan 08 2015: (Start)
%F a(n) = 2 * A000330(n) + A000330(n+1) + A000330(n-1).
%F a(n) = A005900(n) + A005900(n+1).
%F a(n) = A005900(n) + A000330(n) + A000330(n+1).
%F a(n) = A000330(n-1) + A000330(n) + A005900(n+1). (End)
%F a(n) = A002412(n+1) + A016061(n-1) for n > 0. - _Bruce J. Nicholson_, Nov 12 2017
%F E.g.f.: exp(x)*(3 + 18*x + 18*x^2 + 4*x^3)/3. - _Stefano Spezia_, Mar 14 2024
%F Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = 5/6 - log(2) = (1 - 1/2 + 1/3) - log(2). - _Peter Bala_, Mar 21 2024
%t Table[(4 n^3 - 6 n^2 + 8 n - 3)/3, {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 15 2011 *)
%t LinearRecurrence[{4, -6, 4, -1}, {1, 7, 25, 63}, 40] (* _Harvey P. Dale_, Jun 05 2013 *)
%t CoefficientList[Series[(1 + x)^3/(-1 + x)^4, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 27 2017 *)
%o (PARI) a(n)=(2*n+1)*(2*n^2+2*n+3)/3 \\ _Charles R Greathouse IV_, Dec 06 2011
%o (Haskell)
%o a001845 n = (2 * n + 1) * (2 * n ^ 2 + 2 * n + 3) `div` 3
%o -- _Reinhard Zumkeller_, Dec 15 2013
%Y Sums of 2 consecutive terms give A008412.
%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 Partial sums of A005899.
%Y Cf. A001846, A001847, A001848, etc., A014820, A013609.
%Y Cf. also A240876, A002412, A016061, A005899.
%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.
%Y Row/column 3 of A008288.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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