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 A005900 Octahedral numbers: a(n) = n*(2*n^2 + 1)/3. (Formerly M4128) 107
 0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, 19871, 21856, 23969, 26214, 28595, 31116, 33781, 36594, 39559, 42680 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Series reversion of g.f.: A(x) is Sum_{n>0} - A066357(n)(-x)^n. Partial sums of centered square numbers A001844. - Paul Barry, Jun 26 2003 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 Schlaefli symbol for this polyhedron: {3,4}. 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 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 a(n) = largest coefficient of (1 + ... + x^(n-1))^4. - R. H. Hardin, Jul 23 2009 Convolution square root of (1 + 6x + 19x^3 + ...) = (1 + 3x + 5x^2 + 7x^3 + ...) = A005408(x). - Gary W. Adamson, Jul 27 2009 Starting with offset 1 = the triangular series convolved with [1, 3, 4, 4, 4, ...]. - Gary W. Adamson, Jul 28 2009 One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral, and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010 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 Arises in Bezdek's proof on contact numbers for congruent sphere packings (see preprint). - Jonathan Vos Post, Feb 08 2011 Euler transform of length 2 sequence [6, -2]. - Michael Somos, Mar 27 2011 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 a(n) is the number of semistandard Young tableaux over all partitions of 3 with maximal element <= n. - Alois P. Heinz, Mar 22 2012 Self convolution of the odd numbers. - Reinhard Zumkeller, Apr 04 2012 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 The sequence is the third partial sum of (0, 1, 3, 4, 4, 4, ...). - Gary W. Adamson, Sep 11 2015 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 REFERENCES 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. 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 X. Acloque, Polynexus Numbers and other mathematical wonders [broken link] Karoly Bezdek, Contact numbers for congruent sphere packings, arXiv:1102.1198 [math.MG], 2011. Matteo Cavaleri and Alfredo Donno, Some degree and distance-based invariants of wreath products of graphs, arXiv:1805.08989 [math.CO], 2018. Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq (5), m=2. Milan Janjic, Two Enumerative Functions Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019. Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75. Hankyung Ko, Volodymyr Mazorchuk and Rafael Mrđen, Join operation for the Bruhat order and Verma modules, arXiv:2109.01067 [math.RT], 2021. See Remark 5.10 p. 19. A. Lascoux and M.-P. Schützenberger, Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), #R27. T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11). J. K. Merikoski, R. Kumar and  R. A. Rajput, Upper bounds for the largest eigenvalue of a bipartite graph, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013. 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, Octahedral Number. Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA a(n) = 1^2 + 2^2 + ... + (n-1)^2 + n^2 + (n-1)^2 + ... + 2^2 + 1^2. - Amarnath Murthy, May 28 2001 G.f.: x * (1 + x)^2 / (1 - x)^4. a(n) = -a(-n) = (2*n^3 + n) / 3. a(n) = ( ((n+1)^5-n^5) - (n^5-(n-1)^5)) )/30. - Xavier Acloque, Oct 17 2003 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 a(n) = 4*binomial(n,3) + 4*binomial(n,2) + binomial(n,1). - Mitch Harris, Jul 06 2006 a(n) = binomial(n+2,3) + 2*binomial(n+1,3) + binomial(n,3), (this pair generalizes; see A014820, the 4-cross polytope numbers). 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 a(n) = A035597(n)/2. - J. M. Bergot, Jun 11 2012 a(n) = A000578(n) - 2*A000292(n-1) for n>0. - J. M. Bergot, Apr 05 2014 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 E.g.f.: (1/3)*x*(3 + 6*x + 2*x^2)*exp(x). - Ilya Gutkovskiy, Mar 16 2017 a(n) = (A002061(A002061(n+1)) - A002061(A002061(n)))/6. - Daniel Poveda Parrilla, Jun 10 2017 a(n) = 6*a(n-1)/(n-1) + a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018 EXAMPLE G.f. = x + 6*x^2 + 19*x^3 + 44*x^4 + 85*x^5 + 146*x^6 + 231*x^7 + ... MAPLE 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)]; A005900:=(z+1)**2/(z-1)**4; # Simon Plouffe in his 1992 dissertation with(combinat): seq(fibonacci(4, 2*n)/12, n=0..40); # Zerinvary Lajos, Apr 21 2008 MATHEMATICA 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 *) CoefficientList[Series[x (1 + x)^2/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 12 2015 *) PROG (PARI) {a(n) = n*(2*n^2+1)/3}; (PARI) concat(, Vec(x*(1 + x)^2/(1 - x)^4 + O(x^50))) \\ Indranil Ghosh, Mar 16 2017 (Haskell) a005900 n = sum \$ zipWith (*) odds \$ reverse odds             where odds = take n a005408_list a005900_list = scanl (+) 0 a001844_list -- Reinhard Zumkeller, Jun 16 2013, Apr 04 2012 (Maxima) makelist(n*(2*n^2+1)/3, n, 0, 20); /* Martin Ettl, Jan 07 2013 */ (Magma) [n*(2*n^2+1)/3: n in [0..50]]; // Wesley Ivan Hurt, Sep 11 2015 (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 (Python) def a(n): return n*(2*n*n + 1)//3 print([a(n) for n in range(41)]) # Michael S. Branicky, Sep 03 2021 CROSSREFS Sums of 2 consecutive terms give A001845. Cf. A001844. 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. Cf. A022521. Cf. A081277. Row n=3 of A210391. - Alois P. Heinz, Mar 22 2012 Cf. A005408. Cf. A053676, A053677, A053678. Cf. A002061. Cf. A000292 (tetrahedral numbers), A000578 (cubes), A006566 (dodecahedral numbers), A006564 (icosahedral numbers). 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). Sequence in context: A212684 A035495 A061293 * A138357 A299264 A183763 Adjacent sequences:  A005897 A005898 A005899 * A005901 A005902 A005903 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)