cp's OEIS Frontend

A005900 Octahedral numbers: a(n) = n*(2*n^2 + 1)/3.

%I A005900 M4128 #244 Mar 08 2025 09:35:23
%S A005900 0,1,6,19,44,85,146,231,344,489,670,891,1156,1469,1834,2255,2736,3281,
%T A005900 3894,4579,5340,6181,7106,8119,9224,10425,11726,13131,14644,16269,
%U A005900 18010,19871,21856,23969,26214,28595,31116,33781,36594,39559,42680
%N A005900 Octahedral numbers: a(n) = n*(2*n^2 + 1)/3.
%C A005900 Series reversion of g.f.: A(x) is Sum_{n>0} - A066357(n)(-x)^n.
%C A005900 Partial sums of centered square numbers A001844. - _Paul Barry_, Jun 26 2003
%C A005900 Also as a(n) = (1/6)*(4n^3 + 2n), n>0: structured tetragonal diamond numbers (vertex structure 5) (cf. A000447 - structured diamonds); and structured trigonal anti-prism numbers (vertex structure 5) (cf. A100185 - structured anti-prisms). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
%C A005900 Schlaefli symbol for this polyhedron: {3,4}.
%C A005900 If X is an n-set and Y and Z are disjoint 2-subsets of X then a(n-4) is equal to the number of 5-subsets of X intersecting both Y and Z. - _Milan Janjic_, Aug 26 2007
%C A005900 Starting with 1 = binomial transform of [1, 5, 8, 4, 0, 0, 0, ...] where (1, 5, 8, 4) = row 3 of the Chebyshev triangle A081277. - _Gary W. Adamson_, Jul 19 2008
%C A005900 a(n) = largest coefficient of (1 + ... + x^(n-1))^4. - _R. H. Hardin_, Jul 23 2009
%C A005900 Convolution square root of (1 + 6x + 19x^3 + ...) = (1 + 3x + 5x^2 + 7x^3 + ...) = A005408(x). - _Gary W. Adamson_, Jul 27 2009
%C A005900 Starting with offset 1 = the triangular series convolved with [1, 3, 4, 4, 4, ...]. - _Gary W. Adamson_, Jul 28 2009
%C A005900 One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral, and icosahedral) numbers (cf. A053012). - _Daniel Forgues_, May 14 2010
%C A005900 Let b be any product of four different primes. Then the divisor lattice of b^n is of width a(n+1). - _Jean Drabbe_, Oct 13 2010
%C A005900 Arises in Bezdek's proof on contact numbers for congruent sphere packings (see preprint). - _Jonathan Vos Post_, Feb 08 2011
%C A005900 Euler transform of length 2 sequence [6, -2]. - _Michael Somos_, Mar 27 2011
%C A005900 a(n+1) is the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 2n. - _Clark Kimberling_, Mar 19 2012
%C A005900 a(n) is the number of semistandard Young tableaux over all partitions of 3 with maximal element <= n. - _Alois P. Heinz_, Mar 22 2012
%C A005900 Self convolution of the odd numbers. - _Reinhard Zumkeller_, Apr 04 2012
%C A005900 a(n) is the number of (w,x,y,z) with all terms in {1,...,n} and w+x=y+z; also the number of (w,x,y,z) with all terms in {0,...,n} and |w-x|<=y. - _Clark Kimberling_, Jun 02 2012
%C A005900 The sequence is the third partial sum of (0, 1, 3, 4, 4, 4, ...). - _Gary W. Adamson_, Sep 11 2015
%C A005900 a(n) is the number of join-irreducible elements in the Weyl group of type B_n with respect to the strong Bruhat order. - _Rafael Mrden_, Aug 26 2020
%C A005900 Number of unit octahedra contained in an n-scale octahedron composed of a tetrahedral-octahedral honeycomb. The number of unit tetrahedra in it is 8*A000292(n-1) = 4*(n^3 - n)/3. Also, the number of unit tetrahedra and unit octahedra contained in an n-scale tetrahedron composed of a tetrahedral-octahedral honeycomb is respectively A006527(n) = (n^3 + 2*n)/3 and A000292(n-1) = (n^3 - n)/6. - _Jianing Song_, Feb 24 2025
%D A005900 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 50.
%D A005900 H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
%D A005900 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005900 T. D. Noe, <a href="/A005900/b005900.txt">Table of n, a(n) for n = 0..1000</a>
%H A005900 X. Acloque, <a href="http://members.fortunecity.fr/polynexus/index.html">Polynexus Numbers and other mathematical wonders</a> [broken link]
%H A005900 Karoly Bezdek, <a href="http://arxiv.org/abs/1102.1198">Contact numbers for congruent sphere packings</a>, arXiv:1102.1198 [math.MG], 2011.
%H A005900 Matteo Cavaleri and Alfredo Donno, <a href="https://arxiv.org/abs/1805.09368">Some degree and distance-based invariants of wreath products of graphs</a>, arXiv:1805.08989 [math.CO], 2018.
%H A005900 Y-h. Guo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Guo/guo4.html">Some n-Color Compositions</a>, J. Int. Seq. 15 (2012) 12.1.2, eq (5), m=2.
%H A005900 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A005900 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A005900 Muhammad Fatih Killik and Bünyamin Şahi̇n, <a href="https://www.researchgate.net/publication/387461002_Further_Results_On_Level_Matrix">Further Results on Level Matrix</a>, Int'l J. Math. Combin. (2024) Vol 4, 68-74. See p. 71.
%H A005900 Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.
%H A005900 Hankyung Ko, Volodymyr Mazorchuk and Rafael Mrđen, <a href="https://arxiv.org/abs/2109.01067">Join operation for the Bruhat order and Verma modules</a>, arXiv:2109.01067 [math.RT], 2021. See Remark 5.10 p. 19.
%H A005900 A. Lascoux and M.-P. Schützenberger, <a href="http://doi.org/10.37236/1285">Treillis et bases des groupes de Coxeter</a>, Electron. J. Combin. 3 (1996), #R27.
%H A005900 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. (11).
%H A005900 J. K. Merikoski, R. Kumar and  R. A. Rajput, <a href="http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol26_pp168-176.pdf">Upper bounds for the largest eigenvalue of a bipartite graph</a>, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
%H A005900 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 A005900 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005900 B. K. Teo and N. J. A. Sloane, <a href="http://neilsloane.com/doc/magic1/magic1.pdf">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.
%H A005900 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctahedralNumber.html">Octahedral Number</a>.
%H A005900 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A005900 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A005900 a(n) = 1^2 + 2^2 + ... + (n-1)^2 + n^2 + (n-1)^2 + ... + 2^2 + 1^2. - _Amarnath Murthy_, May 28 2001
%F A005900 G.f.: x * (1 + x)^2 / (1 - x)^4. a(n) = -a(-n) = (2*n^3 + n) / 3.
%F A005900 a(n) = ( ((n+1)^5-n^5) - (n^5-(n-1)^5) )/30. - Xavier Acloque, Oct 17 2003
%F A005900 a(n) is the sum of the products pq, where p and q are both positive and odd and p + q = 2n, e.g., a(4) = 7*1 + 5*3 + 3*5 + 1*7 = 44. - _Jon Perry_, May 17 2005
%F A005900 a(n) = 4*binomial(n,3) + 4*binomial(n,2) + binomial(n,1). - _Mitch Harris_, Jul 06 2006
%F A005900 a(n) = binomial(n+2,3) + 2*binomial(n+1,3) + binomial(n,3), (this pair generalizes; see A014820, the 4-cross polytope numbers).
%F A005900 Sum_{n>=1} 1/a(n) = 3*gamma + 3*Psi((I*(1/2))*sqrt(2)) - (1/2)*(3*I)*Pi*coth((1/2)*Pi*sqrt(2)) - (1/2)*(3*I)*sqrt(2) = A175577, where I=sqrt(-1). - _Stephen Crowley_, Jul 14 2009
%F A005900 a(n) = A035597(n)/2. - _J. M. Bergot_, Jun 11 2012
%F A005900 a(n) = A000578(n) - 2*A000292(n-1) for n>0. - _J. M. Bergot_, Apr 05 2014
%F A005900 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - _Wesley Ivan Hurt_, Sep 11 2015
%F A005900 E.g.f.: (1/3)*x*(3 + 6*x + 2*x^2)*exp(x). - _Ilya Gutkovskiy_, Mar 16 2017
%F A005900 a(n) = (A002061(A002061(n+1)) - A002061(A002061(n)))/6. - _Daniel Poveda Parrilla_, Jun 10 2017
%F A005900 a(n) = 6*a(n-1)/(n-1) + a(n-2) for n > 1. - _Seiichi Manyama_, Jun 06 2018
%F A005900 Sum_{n >= 1} (-1)^(n+1)/(a(n)*a(n+1)) = 6*log(2) - 4 = 1/(6 + 2/(6 + 6/(6 + ... + n*(n-1)/(6 + ...)))). See A142983. - _Peter Bala_, Mar 06 2024
%e A005900 G.f. = x + 6*x^2 + 19*x^3 + 44*x^4 + 85*x^5 + 146*x^6 + 231*x^7 + ...
%p A005900 al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(3,n), n=0..100)];
%p A005900 A005900:=(z+1)**2/(z-1)**4; # _Simon Plouffe_ in his 1992 dissertation
%p A005900 with(combinat): seq(fibonacci(4,2*n)/12, n=0..40); # _Zerinvary Lajos_, Apr 21 2008
%t A005900 Table[(2n^3+n)/3, {n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,6,19},50] (* _Harvey P. Dale_, Oct 10 2013 *)
%t A005900 CoefficientList[Series[x (1 + x)^2/(1 - x)^4, {x, 0, 45}], x] (* _Vincenzo Librandi_, Sep 12 2015 *)
%o A005900 (PARI) {a(n) = n*(2*n^2+1)/3};
%o A005900 (PARI) concat([0],Vec(x*(1 + x)^2/(1 - x)^4 + O(x^50))) \\ _Indranil Ghosh_, Mar 16 2017
%o A005900 (Haskell)
%o A005900 a005900 n = sum $ zipWith (*) odds $ reverse odds
%o A005900             where odds = take n a005408_list
%o A005900 a005900_list = scanl (+) 0 a001844_list
%o A005900 -- _Reinhard Zumkeller_, Jun 16 2013, Apr 04 2012
%o A005900 (Maxima) makelist(n*(2*n^2+1)/3, n, 0, 20); /* _Martin Ettl_, Jan 07 2013 */
%o A005900 (Magma) [n*(2*n^2+1)/3: n in [0..50]]; // _Wesley Ivan Hurt_, Sep 11 2015
%o A005900 (Magma) I:=[0,1,6,19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Sep 12 2015
%o A005900 (Python)
%o A005900 def a(n): return n*(2*n*n + 1)//3
%o A005900 print([a(n) for n in range(41)]) # _Michael S. Branicky_, Sep 03 2021
%Y A005900 Sums of 2 consecutive terms give A001845. Cf. A001844.
%Y A005900 1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
%Y A005900 Cf. A022521.
%Y A005900 Cf. A081277.
%Y A005900 Row n=3 of A210391. - _Alois P. Heinz_, Mar 22 2012
%Y A005900 Cf. A005408.
%Y A005900 Cf. A053676, A053677, A053678.
%Y A005900 Cf. A002061.
%Y A005900 Cf. A000292 (tetrahedral numbers), A000578 (cubes), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
%Y A005900 Similar sequence: A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099193(m=7), A099195 (m=8), A099196 (m=9), A099197 (m=10).
%K A005900 nonn,easy
%O A005900 0,3
%A A005900 _N. J. A. Sloane_