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A005900 Octahedral numbers: (2*n^3 + n)/3.
(Formerly M4128)
79
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.

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

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

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

Milan Janjic, Two Enumerative Functions

Hyun Kwang Kim, On Regular Polytope Numbers

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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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 two-way infinite sequences

Index entries for sequences related to 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

Partial sums of centered square numbers A001844. - Paul Barry, Jun 26 2003

Euler transform of length 2 sequence [6, -2]. - Michael Somos, Mar 27 2011

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(1/a(n), n=1..infinity) = 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. - 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

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

f[n_]:=(n+1)^2+n^2; s=0; lst={}; Do[AppendTo[lst, s+=f[n]], {n, 0, 4!}]; lst..and/or.. f[n_]:=(2*n^3+n)/3; lst={}; Do[AppendTo[lst, f[n]], {n, 0, 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 17 2010 *)

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 *)

PROG

(PARI) {a(n) = (2*n^3 + n) / 3}

(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) A005900(n):=(2*n^3 + n)/3$ makelist(A005900(n), n, 0, 20); /* Martin Ettl, Jan 07 2013 */

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.

Sequence in context: A212684 A035495 A061293 * A138357 A183763 A209403

Adjacent sequences:  A005897 A005898 A005899 * A005901 A005902 A005903

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), May 03 2001

Edited (example erased). - Wolfdieter Lang, Apr 19 2014

STATUS

approved

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Last modified July 22 19:50 EDT 2014. Contains 244836 sequences.